of ,mathematics at the ,university of   
,glasgow47                              
----------------------------------------
----------------------------            
,,transcriber',s ,,notes3               
,mathematical expressions which are in  
italics in the original text     are    
displayed in non-italic characters to   
increase the readability     of the     
book4                                   
,for super and subscripts unicode       
characters have been used except      a 
few cases where carat symbol and        
underline have been used4               
,mathematical notation in the body of   
text is duly conserved but              
parantheses have been added to          
expressions to avoid any kind of        
misenterpretation4                      
----------------------------------------
----------------------------            
,,english ,,men ,,of ,,science          
,,edited ,,by   ,j4 ,,reynolds ,,green1 
,sc4,d4    ,,lord ,,kelvin              
,,english ,,men   ,,of ,,science        

,,edited ,,by   ,,dr4 ,j4 ,,reynolds    
,,green4    ,with ,photogravure         
,frontispiece4   ,small ,cr4 #hvo1 #bs4 
#f;d4 net per vol4    ,,spencer4 ,by ,j4
 ,,arthur ,,thompson4   ,,priestley4 ,by
 ,dr4 ,,thorpe1 ,c4,b41 ,f4,r4,s4       
,,flower4 ,by ,prof4 ,r4 ,,lydekker1    
,f4,r4,s4   ,,huxley4 ,by ,prof4        
,,ainsworth ,,davis4   ,,bentham4 ,by   
,b4 ,,daydon ,,jackson1 ,f4,l4,s4       
,,dalton4 ,by ,j4 ,p4 ,,millington1     
,m4,a4    ,j4 ,m4 ,,dent  ,,co4     ,all
 ,rights ,reserved      ,illustration3  
,lord ,kelvin      ,,lord ,,kelvin      
,,an ,,account ,,of ,,his ,,scientific  
,,life ,,and ,,work     ,,by    ,,andrew
 ,,gray   ,,ll4,d41 ,f4,r4,s41          
,v4-,p4,r4,s4,e4    ,,professor ,,of    
,,natural ,,philosophy ,,in ,,the       
,,university ,,of ,,glasgow             
,,published ,,in ,,london ,,by   ,j4 ,m4
 ,,dent  ,,co41 ,,and ,,in ,,new        
,,york ,,by ,e4 ,p4 ,,dutton  ,,co4     
#aijh      ,,richard ,,clay  ,,sons1    
,,limited1   ,,bread ,,street ,,hill1   

,e4,c41 ,,and   ,,bungay1 ,,suffolk4    
,,preface   ,this book makes no claim to
 be a biography of ,lord ,kelvin in the 
usual sense4 ,it is an extension of an  
article which appeared in the ,glasgow  
,herald for ,december #ai1 #aijg1 and   
has been written at the suggestion of   
various friends of ,lord ,kelvin1 in the
 ,university of ,glasgow and elsewhere1 
who had read that article4 ,the aim of  
the volume is to give an account of     
,lord ,kelvin's life of scientific      
activity1 and to explain to the student1
 and to the general reader who takes an 
interest in physical science and its    
applications1 the nature of his         
discoveries4 ,only such a statement of  
biographical facts as seems in harmony  
with this purpose is attempted4 ,but ,i 
have ventured1 as an old pupil and      
assistant of ,lord ,kelvin1 to sketch   
here and there the scene in his         
class-room and laboratory1 and to record
 some of the incidents of his teaching  
and work4  ,i am under obligations to   

the proprietors of the ,glasgow ,herald 
for their freely accorded permission to 
make use of their article1 and to       
,messrs4 ,annan1 photographers1         
,glasgow1 and ,messrs4 ,james           
,mac,lehose  ,sons1 ,glasgow1 for the   
illustrations which are given1 and which
 ,i hope may add to the interest of the 
book4                                   
,a4 ,,gray4    ,the ,university1        
,glasgow1       ,may #bj1 #aijh4        
,,contents     ,,chap4                  
,,page       ,i4 ,,parentage ,,and      
,,early ,,education                     
#a      ,,ii4 ,,classes ,,at ,,the      
,,university ,,of ,,glasgow4 ,,first    
,,scientific           ,,papers         
#ac     ,,iii4 ,,university ,,of        
,,cambridge4 ,,scientific ,,work ,,as   
,,undergraduate   #bc      ,,iv4 ,,the  
,,mathematical ,,theory ,,of            
,,electricity ,,in ,,equilibrium4       
,,electric ,,images4 ,,electric         
,,inversion                       #cc   
,v4 ,,the ,,chair ,,of ,,natural        

,,philosophy ,,at ,,glasgow4            
,,establishment           ,,of ,,the    
,,first ,,physical ,,laboratory         
#fa      ,,vi4 ,,friendship ,,with      
,,stokes ,,and ,,joule4 ,,early ,,work  
,,at ,,glasgow     #gi     ,,vii4 ,,the 
',,account ,,of ,,carnot',s ,,theory    
,,of ,,the ,,motive ,,power ,,of        
,,heat'--,,transition ,,to ,,the        
,,dynamical ,,theory ,,of ,,heat        
#ii    ,,viii4 ,,thermodynamics ,,and   
,,absolute ,,thermometry                
#aad      ,,ix4                         
,,hydrodynamics--,,dynamical ,,theorem  
,,of ,,minimum                          
,,energy--,,vortex ,,motion             
#aec       ,x4 ,,the ,,energy ,,theory  
,,of ,,electrolysis--,,electrical       
,,units--,,electrical ,,oscillations    
#agf      ,,xi4 ,,thomson ,,and         
,,tait',s ',,natural                    
,,philosophy'--,,gyrostatic             
,,action--',,electrostatics ,,and       
,,magnetism'                   #aid     
,,xii4 ,,the ,,age ,,of ,,the ,,earth   

#bbi    ,,xiii4 ,,british ,,association 
,,committee ,,on ,,electrical           
,,standards      #bdd     ,,xiv4 ,,the  
,,baltimore ,,lectures                  
#bed      ,,xv4 ,,speed ,,of ,,telegraph
 ,,signalling--,,laying ,,of ,,submarine
           ,,cables--,,telegraph        
,,instruments--,,navigational           
,,instruments1 ,,compass ,,and          
,,sounding ,,machine                #bfd
     ,,xvi4 ,,lord ,,kelvin ,,in ,,his  
,,class-,,room ,,and ,,laboratory       
#bgi    ,,xvii4 ,,practical             
,,activities--,,honours ,,and           
,,distinctions--,,last                  
,,illness ,,and ,,death                 
#bii          ,,conclusion              
#cje          ,,index                   
#cag   ,,corrigendum    ,page #aje1 line
 #i from foot1 for e  ,o read e  o      
,,list ,,of ,,illustrations             
,to face page   ,,lord ,,kelvin         
7photogravure7                          
,frontispiece   ,,lord ,,kelvin ,,in    
#ahdf                                   

#fd   ,,view ,,of ,,old ,,college       
#gj     ,,lord ,,kelvin     ,,chapter ,i
  ,,parentage ,,and ,,early ,,education 
,lord ,kelvin came of a stock which has 
helped to give to the north of ,ireland 
its commercial and industrial supremacy 
over the rest of that distressful       
country4 ,his ancestors were county     
,down agriculturists of ,scottish       
extraction4 ,his father was ,james      
,thomson1 the well-known ,glasgow       
,professor of ,mathematics1 and author  
of mathematical text-books which at one 
time were much valued1 and are even now 
worth consulting4 ,james ,thomson was   
born on ,november #ac1 #aghf1 near      
,ballynahinch1 county ,down4 ,being the 
son of a small farmer he was probably   
unable to enter on university studies at
 the usual age1 for he did not          
matriculate in ,scotland until #ahaj4   
,the class-lists of the time show that  
he distinguished himself highly in      
mathematics1 natural philosophy1 and    
classics4  ,an interesting incident of  

these student days of his father was    
related by ,lord ,kelvin in his         
installation address as ,chancellor of  
the ,university in #aijd1 and is        
noteworthy as indicating how            
comparatively recent are many of the    
characteristics of our present-day life 
and commerce4 ,james ,thomson and some  
companions1 walking from ,greenock to   
,glasgow1 on their way to join the      
college classes at the commencement of  
the session1 saw a prodigy--a black     
chimney moving rapidly beyond a field on
 the left-hand side of their road4 ,they
 jumped the fence1 ran across the field1
 and saw1 to their astonishment1 ,henry 
,bell's ',comet' 7then not a year old7  
travelling on the ,clyde between        
,glasgow and ,greenock4#a ,sometimes    
then the passage from ,belfast to       
,greenock took a long time4 ,once ,james
 ,thomson1 crossing in an old           
lime-carrying smack1 was three or four  
days on the way1 in the course of which 
the vessel1 becalmed1 was carried three 

times by the tide round ,ailsa ,craig4  
,mr4 ,thomson was elected in #ahae to   
the ,professorship of ,mathematics in   
the ,royal ,academical ,institution of  
,belfast1 and held the post for         
seventeen years1 building up for himself
 an excellent reputation as a teacher1  
and as a clear and accurate writer4     
,just then analytical methods were      
beginning to supersede the processes of 
geometrical demonstration which the form
 adopted by ,newton for the ,principia  
had tended to perpetuate in this        
country4 ,laplace was at the height of  
his fame in ,france1 and was writing the
 great analytical ,principia1 his       
,mcanique ,cleste1 applying the whole   
force of his genius1 and all the        
resources of the differential and       
integral calculus invented by ,newton   
and improved by the mathematicians of   
the intervening century1 to the         
elucidation and extension of the system 
of the world1 which had been so boldly  
sketched by the founder of modern       

physical science4  ,in that period      
,fourier wrote his memoirs on the       
conduction of heat1 and gave to the     
world his immortal book to be an        
inspiration to the physical philosophers
 of succeeding generations4 ,legendre   
had written memoirs which were to lead1 
in the hands of ,jacobi and his         
successors1 to a new province of        
mathematics1 while1 in ,germany1 ,gauss 
had begun his stately march of          
discovery4  ,the methods and results of 
this period of mathematical activity    
were at first hardly known in this      
country3 the slavish devotion of        
,cambridge to the geometrical processes 
and the fluxional notation of ,newton1  
an exclusive partiality which ,newton   
himself would have been the first to    
condemn1 led analytical methods1 equally
 ,newtonian1 to be stigmatised as       
innovations1 because clothed in the     
unfamiliar garb of the continental      
notation4 ,a revolt against this was led
 by ,sir ,john ,herschel1 ,woodhouse1   

,peacock1 and some others at ,cambridge1
 who wrote books which had a great      
effect in bringing about a change of    
methods4 ,sir ,john thus described the  
effect of the new movements3--,students 
at our universities1 fettered by no     
prejudices1 entangled by no habits1 and 
excited by the ardour and emulation of  
youth1 had heard of the existence of    
masses of knowledge from which they were
 debarred by the mere accident of       
position4 ,they required no more4 ,the  
prestige which magnifies what is        
unknown1 and the attractions inherent in
 what is forbidden1 coincided in their  
impulse4 ,the books were procured and   
read1 and produced their natural        
effects4 ,the brows of many a ,cambridge
 examiner were elevated1 half in ire1   
half in admiration1 at the unusual      
answers which began to appear in        
examination papers4 ,even moderators are
 not made of impenetrable stuff1 though 
fenced with sevenfold ,jacquier1 and    
tough bull-hide of ,vince and ,wood4    

,the memoirs and treatises of the       
continental analysts were eagerly       
procured and studied by ,james ,thomson1
 and as he was bound by no examination  
traditions1 he freely adopted their     
methods1 so far as these came within the
 scope of his teaching1 and made them   
known to the ,english reading public in 
his text-books4 ,hence when the chair of
 ,mathematics at ,glasgow became vacant 
in #ahcb by the death of ,mr4 ,james    
,millar1 ,mr4 ,thomson was at once      
chosen by the ,faculty1 which at that   
time was the electing body4  ,the       
,faculty consisted of the ,principal and
 the ,professors of ,divinity1 ,church  
,history1 ,oriental ,languages1 ,natural
 ,philosophy1 ,moral ,philosophy1       
,mathematics1 ,logic1 ,greek1 ,humanity1
 ,civil ,law1 ,practice of ,medicine1   
,anatomy1 and ,practical ,astronomy4 ,it
 administered the whole revenues and    
property of the ,college1 and possessed 
the patronage of the above-named chairs 
with the exception of ,church ,history1 

,civil ,law1 ,medicine1 ,anatomy1 and   
,astronomy1 so that ,mr4 ,thomson became
 not only ,professor of ,mathematics1   
but also1 in virtue of his office1 a    
member of what was really the supreme   
governing body of the ,university4 ,the 
members of the ,faculty1 with the       
exception of the ,professor of          
,astronomy1 who resided at the          
observatory1 were provided with official
 residences in the ,college4 ,this      
arrangement is still adhered to2 though 
now the government is in the hands of a 
,university ,court1 with the ,senate    
7which formerly only met to confer      
degrees or to manage the library and    
some other matters7 to regulate and     
superintend teaching and discipline4    
,professor ,thomson was by no means the 
first or the only professor of the name 
in the ,university of ,glasgow1 as the  
following passage quoted from a letter  
of ,john ,nichol1 son of ,dr4 ,j4 ,p4   
,nichol1 and first ,professor of        
,english at ,glasgow1 amusingly         

testifies3--  ,niebuhr1 after examining 
a portion of the ,fasti ,consulares1    
arrived at the conclusion that the      
senatus populusque ,romanus had made a  
compact to elect every year a member of 
the ,fabian house to one of the highest 
offices of state1 so thickly are the    
records studded with the name of the    
,fabii4 ,some future ,niebuhr of the    
,new ,zealand ,macaulay imagines1       
turning his attention to the annals of  
,glasgow ,college1 will undoubtedly     
arrive at the conclusion that the       
leaders of that illustrious corporation 
had1 during the period of which ,i am   
writing1 become bound in a similar      
manner to the name of ,thomson4 ,members
 of that great gens filled one-half of  
the chairs in the ,university4 ,i will  
not venture to say how many ,i have     
known4 ,there was ,tommy ,thomson the   
chemist2 ,william ,thomson of ,materia  
,medica2 ,allen ,thomson of ,anatomy1   
brother of the last2 ,dr4 ,james        
,thomson of ,mathematics2 ,william1 his 

son1 etc41 etc4 ,old ,dr4 ,james was one
 of the best of ,irishmen1 a good       
mathematician1 an enthusiastic and      
successful teacher1 the author of       
several valuable school-books1 a friend 
of my father's1 and himself the father  
of a large family1 the members of which 
have been prosperous in the world4 ,they
 lived near us in the court1 and we made
 a pretty close acquaintanceship with   
them all4  ,a former ,professor of      
,natural ,philosophy1 ,dr4 ,anderson1#b 
who appears to have lived the closing   
years of his life in almost constant    
warfare with his colleagues of the      
,faculty1 and who established science   
classes for workmen in ,glasgow1        
bequeathed a sum of money to set up a   
college in ,glasgow in which such       
classes might be carried on4 ,the result
 was the foundation of what used to be  
called the ,andersonian ,university in  
,george ,street1 the precursor of the   
magnificent ,technical ,college of the  
present day4 ,this name1 and the large  

number of ,thomsons who had been and    
were still connected with the           
,university of ,glasgow1 caused the more
 ancient institution to be not          
infrequently referred to as the         
,thomsonian ,university6  ,the ,thomas  
,thomson 7no relative of the ,belfast   
,thomsons7 affectionately1 if a little  
irreverently1 mentioned in the above    
quotation1 was then the ,professor of   
,chemistry4 ,he was the first to        
establish a chemical laboratory for     
students in this country2 indeed1 his   
laboratory preceded that of ,liebig at  
,giessen by some years1 and it is       
probable that as regards experimental   
chemistry ,glasgow was then in advance  
of the rest of the world4 ,his pupil and
 life-long admirer was destined to      
establish the first physical laboratory 
for such students as were willing to    
spend some time in the experimental     
investigation and verification of       
physical principles1 or to help the     
professor in his researches4 ,the       

systematic instruction of students in   
methods of experimenting by practical   
exercises with apparatus was a much     
later idea1 and this fact must be taken 
account of when the laboratories of the 
present time are contrasted with the    
much more meagre provision of those     
early days4 ,the laboratory is now1 as  
much as the lecture-room1 the place     
where classes are held and instruction  
given in experimental science to crowds 
of students1 and it is a change for the 
better4  ,the arrival of ,james ,thomson
 and his family at ,glasgow ,college1 in
 #ahcb1 was remarked at the time as an  
event which brought a large             
reinforcement to the gens already       
inseparably associated with the place3  
how great were to be its consequences   
not merely to the ,university but to the
 world at large nobody can then have    
imagined4 ,his family consisted of four 
sons and two daughters3 his wife1       
,margaret ,gardner1 daughter of ,william
 ,gardner1 a merchant in ,glasgow1 had  

died shortly before1 and the care of the
 family was undertaken by her sister1   
,mrs4 ,gall4 ,the eldest son1 ,james    
,thomson1 long after to be ,rankine's   
successor in the ,chair of ,engineering1
 was ten years of age and even then an  
inveterate inventor2 ,william1 the      
future ,lord ,kelvin 7born ,june #bf1   
#ahbd71 was a child of eight4 ,two      
younger sons were ,john 7born in        
#ahbf7--who achieved distinction in     
,medicine1 became ,resident ,assistant  
in the ,glasgow ,royal ,infirmary1 and  
died there of a fever caught in the     
discharge of his duty--and ,robert1 who 
was born in #ahbi1 and died in          
,australia in #aije4 ,besides these four
 sons there were in all three           
daughters3--,elizabeth1 afterwards wife 
of the ,rev4 ,david ,king1 ,d4,d42      
,anna1 who was married to ,mr4 ,william 
,bottomley of ,belfast 7these two were  
the eldest of the family71 and          
,margaret1 the youngest1 who died in    
childhood4 ,thus began ,william         

,thomson's residence in and connection  
with the ,university of ,glasgow1 a     
connection only terminated by the       
funeral ceremony in ,westminster ,abbey 
on ,december #bc1 #aijg4  ,professor    
,thomson himself carefully superintended
 the education of his sons1 which was   
carried out at home4 ,they were well    
grounded in the old classical languages1
 and moreover received sound instruction
 in what even now are called1 but in a  
somewhat disparaging sense1 modern      
subjects4 ,as ,john ,nichol has said in 
his letters1 ,he was a stern            
disciplinarian1 and did not relax his   
discipline when he applied it to his    
children1 and yet the aim of his life   
was their advancement4  ,it would appear
 from ,john ,nichol's recollections that
 even in childhood and youth1 young     
,james ,thomson was an enthusiastic     
experimentalist and inventor1 eager to  
describe his ideas and show his models  
to a sympathetic listener4#c ,and both  
then and in later years his charming    

simplicity1 his devouring passion for   
accuracy of verbal expression in all his
 scientific writing and teaching1 and   
his unaffected and unconscious genius   
for the invention of mechanical         
appliances1 all based on true and       
intuitively perceived physical          
principles1 showed that if he had had   
the unrelenting power of ignoring       
accessories and unimportant details     
which was possessed by his younger      
brother1 he might have accomplished far 
more than he did1 considerable as that  
was4 ,but ,william had more rapid       
decision1 and though careful and exact  
in expressing his meaning1 was less     
influenced by considerations of the     
errors that might arise from the various
 connotations of such scientific terms  
as are also words in common use2 and he 
quickly completed work which his brother
 would have pondered over for a long    
time1 and perhaps never finished4  ,it  
is difficult for a stranger to ,glasgow1
 or even for a resident in ,glasgow in  

these days of quick and frequent        
communication with ,england1 and for    
that matter with all parts of the world1
 to form a true idea of life and work at
 the ,university of ,glasgow seventy    
years ago4 ,the ,university had then its
 home in the old tounis colledge in the 
,high ,street1 where many could have    
wished it to remain1 and1 extending its 
buildings on ,college ,green1 retain the
 old and include the new4 ,its fine old 
gateway1 and part of one of the courts1 
were still a quaint adornment of the    
somewhat squalid street in #ahga1 after 
the ,university had moved to its present
 situation on the windy top of          
,gilmorehill4 ,deserted as it was1 its  
old walls told something of the history 
of the past1 and reminded the passer-by 
that learning had flourished amid the   
shops and booths of the townspeople1 and
 that students and professors had there 
lived and worked within sound of the    
shuttle and the forge4 ,the old         
associations of a town or a street or a 

building1 linked as they often are with 
the history of a nation1 are a valuable 
possession1 not always placed in the    
account when the advantages or          
disadvantages of proposed changes are   
discussed2 but a ,university which for  
four hundred years has seen the tide of 
human life flow round it in a great     
city1 is instinct with memories which   
even the demolition of its walls can    
only partially destroy4 ,poets and      
statesmen1 men of thought and men of    
action1 lords and commoners1 rich men's 
sons and the children of farmers1       
craftsmen and labourers1 had mingled in 
its classes and sat together on its     
benches2 and so had been brought about a
 community of thought and feeling which 
the practice of our modern and wealthy  
cosmopolites1 who affect to despise     
nationality1 certainly does nothing to  
encourage4 ,in the eighteenth century   
the ,provosts and the ,bailies of the   
time still dwelt among men and women in 
the ,high ,street1 and its continuation 

the ,saltmarket1 or not far off in      
,virginia ,street1 the home of the      
tobacco lords and the ,west ,india      
merchants4 ,their homely hospitality1   
their cautious and at the same time     
splendid generosity1 their prudent      
courage1 and their faithful and candid  
friendships are depicted in the pages of
 ,scott2 and though a change in men and 
manners1 not altogether for the better1 
has been gradually brought about by     
sport and fashion1 those peculiarly     
,scottish virtues are still to be found 
in the civic statesmen and merchant     
princes of the ,glasgow of to-day4      
,seventy years ago the great migration  
of the well-to-do towards the west had  
commenced1 but it had but little        
interfered with the life of the ,high   
,street or of the ,college4 ,now many   
old slums besides the ,vennel and the   
,havannah have disappeared1 much to the 
credit of the ,corporation of ,glasgow2 
and1 alas1 so has every vestige of the  
,old ,college1 much to the regret of all

 who remember its quaint old courts4 ,a 
railway company1 it is to be supposed1  
dare not possess an artistic soul to be 
saved2 and therefore1 perhaps1 it is    
that it builds huge and ugly            
caravanserais of which no one1 except   
perhaps the shareholders1 would keenly  
regret the disappearance4 ,but both     
artists and antiquaries would have      
blessed the directors--and such a       
blessing would have done them no        
harm--if they had been ingenious and    
pious enough to leave some relic of the 
old buildings as a memorial of the old  
days and the old life of the ,high      
,street4  ,a picture of the ,college in 
the ,high ,street has recently been     
drawn by one who lived and worked in it1
 though some thirty years after ,james  
,thomson brought his family to live in  
its courts4 ,professor ,g4 ,g4 ,ramsay  
has thus portrayed some features of the 
place1 which may interest those who     
would like to imagine the environment in
 which ,lord ,kelvin grew up from       

childhood1 until1 a youth of seventeen1 
he left ,glasgow for ,cambridge4#d      
,there was something in the very        
disamenities of the old place that      
created a bond of fellowship among those
 who lived and worked there1 and that   
makes all old students1 to this day1    
look back to it with a sort of family   
pride and reverence4 ,the grimy1 dingy1 
low-roofed rooms2 the narrow1           
picturesque courts1 buzzing with        
student-life2 the dismal1 foggy mornings
 and the perpetual gas2 the sudden      
passage from the brawling1 huckstering  
,high ,street into the academic         
quietude1 or the still more academic    
hubbub1 of those quaint cloisters1 into 
which the policeman1 so busy outside1   
was never permitted to penetrate2 the   
tinkling of the 'angry bell' that made  
the students hurry along to the door    
which was closed the moment that it     
stopped2 the roar and the flare of the  
,saturday nights1 with the cries of     
carouse or incipient murder which would 

rise into our quiet rooms from the      
,vennel or the ,havannah2 the exhausted 
lassitude of ,sunday mornings1 when poor
 slipshod creatures might be seen1 as   
soon as the street was clear of         
churchgoers1 sneaking over to the       
chemist's for a dose of laudanum to ease
 off the debauch of yesterday2 the      
conversations one would have after      
breakfast with the old ladies on the    
other side of the ,vennel1 not twenty   
feet from one's breakfast-table1 who    
divided the day between smoking short   
cutty pipes and drinking poisonous black
 tea--these sharp contrasts bound       
together the ,college folk and the      
,college students1 making them feel at  
once part of the veritable populace of  
the city1 and also hedged off from it by
 separate pursuits and interests4  ,the 
university removed in #ahga to larger   
and more airily situated buildings in   
the western part of the city4 ,round    
these have grown up1 in the intervening 
thirty-eight years1 new buildings for   

most of the great departments of        
science1 including a separate ,institute
 of ,natural ,philosophy1 which was     
opened in ,april #aijg1 by the ,prince  
and ,princess of ,wales4     ,,chapter  
,,ii  ,,classes ,,at ,,the ,,university 
,,of ,,glasgow4 ,,first ,,scientific    
,,papers   ,in #ahcd1 that is at the age
 of ten1 ,william ,thomson entered the  
,university classes4 ,though small in   
stature1 and youthful even for a time   
when mere boys were ,university         
students1 he soon made himself          
conspicuous by his readiness in         
answering questions1 and by his general 
proficiency1 especially in mathematical 
and physical studies4 ,the classes met  
at that time twice a day--in mathematics
 once for lecture and once for oral     
examination and the working of unseen   
examples by students of the class4 ,it  
is still matter of tradition how1 in his
 father's class1 ,william was           
conspicuous for the brilliancy of the   
work he did in this second hour4 ,his   

elder brother ,james and he seem to have
 gone through their ,university course  
together4 ,in #ahcd-e they were         
bracketed third in ,latin ,prose        
,composition4 ,in #ahce-f ,william      
received a prize for a vacation         
exercise--a translation of ,lucian's    
,dialogues of the ,gods with full       
parsing of the first three ,dialogues4  
,in #ahcf-g and #ahcg-h the brothers    
were in the ,junior and ,senior         
,mathematical ,classes1 and in each year
 the first and the second place in the  
prize-list fell to ,william and ,james  
respectively4 ,in the second of these   
years1 ,william appears as second       
prizeman in the ,logic ,class1 while    
,james was third1 and ,john ,caird      
7afterwards ,principal of the           
,university7 was fifth4 ,william and    
,james ,thomson took the first and      
second prizes in the ,natural           
,philosophy ,class at the close of      
session #ahch-i2 and in that year       
,william gained the ,class ,prize in    

,astronomy1 and a ,university ,medal for
 an ,essay on the ,figure of the ,earth4
 ,in #ahdj-a he appears once more1 this 
time as fifth prizeman in the ,senior   
,humanity ,class4  ,in his inaugural    
address as ,chancellor of the           
,university1 already quoted above1 ,lord
 ,kelvin refers to his teachers in      
,glasgow ,college in the following      
words3  ,to this day ,i look back to    
,william ,ramsay's lectures on ,roman   
,antiquities1 and readings of ,juvenal  
and ,plautus1 as more interesting than  
many a good stage play that ,i have seen
 in the theatre4444  ,greek under ,sir  
,daniel ,sandford and ,lushington1      
,logic under ,robert ,buchanan1 ,moral  
,philosophy under ,william ,fleming1    
,natural ,philosophy and ,astronomy     
under ,john ,pringle ,nichol1 ,chemistry
 under ,thomas ,thomson1 a very advanced
 teacher and investigator1 ,natural     
,history under ,william ,cowper1 were1  
as ,i can testify by my experience1 all 
made interesting and valuable to the    

students of ,glasgow ,university in the 
thirties and forties of the nineteenth  
century4444  ,my predecessor in the     
,natural ,philosophy chair1 ,dr4        
,meikleham1 taught his students         
reverence for the great ,french         
mathematicians ,legendre1 ,lagrange1 and
 ,laplace4 ,his immediate successor in  
the teaching of the ,natural ,philosophy
 ,class1#e ,dr4 ,nichol1 added ,fresnel 
and ,fourier to this list of scientific 
nobles3 and by his own inspiring        
enthusiasm for the great ,french school 
of mathematical physics1 continually    
manifested in his experimental and      
theoretical teaching of the wave theory 
of light and of practical astronomy1 he 
largely promoted scientific study and   
thorough appreciation of science in the 
,university of ,glasgow4444  ,as far    
back as #ahah to #ahcj ,thomas ,thomson1
 the first ,professor of ,chemistry in  
the ,university of ,glasgow1 began the  
systematic teaching of practical        
chemistry to students1 and1 aided by the

 ,faculty of ,glasgow ,college1 which   
gave the site and the money for the     
building1 realised a well-equipped      
laboratory1 which preceded1 ,i believe1 
by some years ,liebig's famous          
laboratory of ,giessen1 and was1 ,i     
believe1 the first established of all   
the laboratories in the world for       
chemical research and the practical     
instruction of ,university students in  
chemistry4 ,that was at a time when an  
imperfectly informed public used to     
regard the ,university of ,glasgow as a 
stagnant survival of medivalism1 and    
used to call its professors the ',monks 
of the ,molendinar'6  ,the ,university  
of ,adam ,smith1 ,james ,watt1 and      
,thomas ,reid was never stagnant4 ,for  
two centuries and a half it has been    
very progressive4 ,nearly two centuries 
ago it had a laboratory of human        
anatomy4 ,seventy-five years ago it had 
the first chemical students' laboratory4
 ,sixty-five years ago it had the first 
,professorship of ,engineering of the   

,british ,empire4 ,fifty years ago it   
had the first physical students'        
laboratory--a deserted wine-cellar of an
 old professorial house1 enlarged a few 
years later by the annexation of a      
deserted examination-room4 ,thirty-four 
years ago1 when it migrated from its    
four-hundred-years-old site off the     
,high ,street of ,glasgow to this       
brighter and airier hill-top1 it        
acquired laboratories of physiology and 
zoology2 but too small and too meagrely 
equipped4  ,in the summer of #ahdj      
,professor ,james ,thomson and his two  
sons went for a tour in ,germany4 ,it   
was stipulated that ,german should be   
the chief1 if not the only1 subject of  
study during the holidays4 ,but ,william
 had just begun to study ,fourier's     
famous book1 ,la ,thorie ,analytique de 
la ,chaleur1 and took it with him4 ,he  
read that great work1 full as it was of 
new theorems and processes of           
mathematics1 with the greatest delight1 
and finished it in a fortnight4 ,the    

result was his first original paper ,on 
,fourier's ,expansions of ,functions in 
,trigonometrical ,series1 which is dated
 ,frankfort1 ,july #ahdj1 and ,glasgow1 
,april #ahda1 and was published in the  
,cambridge ,mathematical ,journal 7vol4 
ii1 ,may #ahda74 ,the object of the     
paper is to show in what cases a        
function f7x71 which is to have certain 
arbitrary values between certain values 
of x1 can be expanded in a series of    
sines and when in a series of cosines4  
,the conclusion come to is that1 for    
assigned limits of x1 between #j and a1 
say1 and for the assigned values of the 
function1 f7x7 can be expressed either  
as a series of sines or as a series of  
cosines4 ,if1 however1 the function is  
to be calculated for any value of x1    
which lies outside the limits of that   
variable between which the values of the
 function are assigned1 the values of   
f7x7 there are to be found from the     
expansion adopted1 by rules which are   
laid down in the paper4  ,fourier used  

sine-expansions or cosine-expansions as 
it suited him for the function between  
the limits1 and his results had been    
pronounced to be nearly all erroneous4  
,from this charge of error1 which was   
brought by a distinguished and          
experienced mathematician1 the young    
analyst of sixteen successfully         
vindicated ,fourier's work4 ,fourier was
 incontestably right in holding1 though 
he nowhere directly proved1 that a      
function given for any value of x       
between certain limits1 could be        
expressed either by a sine-series or by 
a cosine-series4 ,the divergence of the 
values of the two expressions takes     
place outside these limits1 as has been 
stated above4  ,the next paper is of the
 same final date1 but appeared in the   
,cambridge ,mathematical ,journal of the
 following ,november4 ,in his treatment 
of the problem of the cooling of a      
sphere1 given with an arbitrary initial 
distribution of temperature symmetrical 
about the centre1 ,fourier assumes that 

the arbitrary function ,f7x71 which     
expresses the temperature at distance x 
from the centre1 can be expanded in an  
infinite series of the form    a sin nx 
a sin nx  444  where a1 a1 444 are      
multipliers to be determined and n1 n1  
444 are the roots1 infinite in number1  
of the transcendental equation 7tan     
n,x7n,x  #a  h,x4  ,this equation       
expresses1 according to a particular    
solution of the differential equation of
 the flow of heat in the sphere1 the    
condition fulfilled at the surface1 that
 the heat reaching the surface by       
conduction from the interior in any time
 is radiated in that time to the        
surroundings4 ,thomson dealt in this    
second paper with the possibility of the
 expansion4 ,he showed that1 inasmuch as
 the first of the roots of the          
transcendental equation lies between #j 
and #ab1 the second between #a and #cb1 
the third between #b and #eb1 and so on1
 with very close approach to the upper  
limit as the roots become of high order1

 the series assumed as possible has     
between the given limits of x the same  
value as the series    ,a sin 7#ab7x  ,a
 sin 7#cb7x  444  where ,a1 ,a1 444 are 
known in terms of a1 a1 444 ,conversely1
 any series of this form is capable of  
being replaced by a series of the form  
assumed4 ,further1 a series of the form 
just written can be made to represent   
any arbitrary system of values between  
the given limits1 and so the possibility
 of the expansion is demonstrated4  ,the
 next ten papers1 with two exceptions1  
are all on the motion of heat1 and      
appeared in the ,cambridge ,mathematical
 ,journal between #ahda and #ahdc1 and  
deal with important topics suggested by 
,fourier's treatise4 ,of the ideas      
contained in one or two of them some    
account will be given presently4        
,fourier's book was called by ,clerk    
,maxwell1 himself a man of much         
spirituality of feeling1 and no mean    
poet1 a great mathematical poem4        
,thomson often referred to it in similar

 terms4 ,the idea of the mathematician  
as poet may seem strange to some2 but   
the genius of the greatest              
mathematicians is akin to that of the   
true creative artist1 who is veritably  
inspired4 ,for such a book was a work of
 the imagination as well as of the      
reason4 ,it contained a new method of   
analysis applied with sublime success to
 the solution of the equations of heat  
conduction1 an analysis which has since 
been transferred to other branches of   
physical mathematics1 and has           
illuminated them with just those rays   
which could reveal the texture and      
structure of the physical phenomena4    
,that method and its applications came  
from ,fourier's mind in full            
development2 he trod unerringly in its  
use along an almost unknown path1 with  
pitfalls on every side2 and he reached  
results which have since been verified  
by a criticism searching and keen1 and  
lasting from ,fourier's day to ours4    
,the criticism has been minute and      

logical3 it has not1 it is needless to  
say1 been poetical4  ,two other great   
works of his father's collection of     
mathematical books1 ,laplace's ,mcanique
 ,cleste and ,lagrange's ,mcanique      
,analytique1 seem also to have been read
 about this time1 and to have made a    
deep impression on the mind of the      
youthful philosopher4 ,the effect of    
these books can be easily traced in     
,thomson and ,tail's ,natural           
,philosophy4  ,the study of ,fourier had
 a profound influence on ,thomson's     
future work1 an influence which has     
extended to his latest writings on the  
theory of certain kinds of waves4 ,his  
treatment is founded on a strikingly    
original use of a peculiar form of      
solution 7given by ,fourier7 of a       
certain fundamental differential        
equation in the theory of the flow of   
heat4 ,it is probable that ,william     
,thomson's earliest predilections as    
regards study were in the direction of  
mathematics rather than of physics4 ,but

 the studies of the young mathematician1
 for such in a very real and high sense 
he had become1 were widened and deepened
 by the interest in physical things and 
their explanation aroused by the        
lectures of ,meikleham1 then ,professor 
of ,natural ,philosophy1 and especially 
7as ,lord ,kelvin testified in his      
inaugural address as ,chancellor7 by the
 teaching of ,j4 ,p4 ,nichol1 the       
,professor of ,astronomy1 a man of      
poetical imagination and of great gifts 
of vivid and clear exposition4  ,the    
,cyclopdia of ,physical ,science which  
,dr4 ,nichol published is little known  
now2 but the first edition1 published in
 #aheg1 to which ,thomson contributed   
several articles1 including a sketch of 
thermodynamics1 contained much that was 
new and stimulating to the student of   
natural philosophy1 and some idea of the
 accomplishments of its compiler and    
author can be gathered from its perusal4
 ,de ,morgan's ,differential and        
,integral ,calculus was a favourite book

 in ,thomson's student days1 and later  
when he was at ,cambridge1 and he       
delighted to pore over its pages before 
the fire when the work of the day was   
over4 ,long after1 he paid a grateful   
tribute to ,de ,morgan and his great    
work1 in the ,presidential ,address to  
the ,british ,association at its        
,edinburgh ,meeting in #ahgj4  ,the next
 paper which ,thomson published1 after  
the two of which a sketch has been given
 above1 was entitled ,the ,uniform      
,motion of ,heat in ,homogeneous ,solid 
,bodies1 and its ,connection with the   
,mathematical ,theory of ,electricity4  
,it is dated ,lamlash1 ,august #ahda1 so
 that it followed the first two at an   
interval of only four months4 ,it       
appeared in the ,cambridge ,mathematical
 ,journal in ,february #ahdb1 and is    
republished in the ,reprint of ,papers  
on ,electrostatics and ,magnetism4 ,it  
will always be a noteworthy paper in the
 history of physical mathematics4 ,for  
although1 for the most part1 only known 

theorems regarding the conduction of    
heat were discussed1 an analogy was     
pointed out between the distribution of 
lines of flow and surfaces of equal     
temperature in a solid and unequally    
heated body1 with sources of heat in its
 interior1 and the arrangement of lines 
of forces and equipotential surfaces in 
an insulating medium surrounding        
electrified bodies1 which correspond to 
the sources of heat in the thermal case4
 ,the distribution of lines of force in 
a space filled with insulating media of 
different inductive qualities was shown 
to be precisely analogous to that of    
lines of flow of heat in a corresponding
 arrangement of media of different      
heat-conducting powers4 ,so the whole   
analysis and system of solutions in the 
thermal case could be at once           
transferred to the electrical one4 ,the 
idea of the conduction of lines of      
force1 as ,faraday first and ,thomson   
afterwards called it1 was further       
developed in subsequent papers1 and     

threw light on the whole subject of     
electrostatic force in the field        
surrounding an electric distribution4   
,moreover1 it made the subject definite 
and quantitative1 and not only gave a   
guide to the interpretation of          
unexplained facts1 but opened a way to  
new theorems and to further             
investigation4  ,this paper contains the
 extremely important theorem of the     
equivalence1 so far as external field is
 concerned1 of any distribution of      
electricity and a certain definite      
distribution1 over any equipotential    
surface1 of a quantity equal to that    
contained within the surface4 ,but this 
general theorem and others contained in 
the paper had been anticipated in       
,green's ,essay on the ,application of  
,mathematical ,analysis to the ,theories
 of ,electricity and ,magnetism1 in     
memoirs by ,chasles in ,liouville's     
,journal 7vols4 iii and v71 and in the  
celebrated memoir by ,gauss ,on ,general
 ,theorems relating to ,attractive and  

,repulsive ,forces varying inversely as 
the ,square of the ,distance1 published 
in ,german in ,leipzig in #ahdj1 and in 
,english in ,taylor's ,scientific       
,memoirs in #ahdb4 ,these anticipations 
are again referred to below4            
,,chapter ,,iii  ,,university ,,of      
,,cambridge4 ,,scientific ,,work ,,as   
,,undergraduate   ,thomson entered at   
,st4 ,peter's ,college1 ,cambridge1 in  
,october #ahda1 and began the course of 
study then in vogue for mathematical    
honours4 ,at that time1 as always down  
almost to the present day1 everything   
depended on the choice of a private     
tutor or coach1 and the devotion of the 
pupil to his directions1 and on         
adherence to the subjects of the        
programme4 ,his private tutor was       
,william ,hopkins1 best of all private  
tutors1 one of the most eminent of his  
pupils called him1 a man of great       
attainment and of distinction as an     
original investigator in a subject which
 had always deeply interested           

,thomson--the internal rigidity of the  
earth4 ,but the curriculum for the      
tripos did not exhaust ,thomson's       
energy1 nor was it possible to keep him 
entirely to the groove of mastering and 
writing out book-work1 and to the       
solution of problems of the kind dear to
 the heart of the mathematical examiner4
 ,he wrote original articles for the    
,cambridge ,mathematical ,journal1 on   
points in pure and in applied           
mathematics1 and read mathematical books
 altogether outside the scope of the    
tripos4 ,nor did he neglect athletic    
exercises and amusements2 he won the    
,colquhoun ,sculls as an oarsman1 and   
was an active member1 and later1 during 
his residence at ,cambridge1 president  
of the ,c4,u4,m4,s41 the ,cambridge     
,university ,musical ,society4#f ,the   
musical instruments he favoured were the
 cornet and especially the ,french      
horn--he was second horn in the original
 ,peterhouse band--but nothing seems to 
be on record as to the difficulties or  

incidents of his practice6 ,long        
afterwards1 in a few extremely          
interesting lectures which he gave      
annually on sound1 he discoursed on the 
vibrations of columns of air in wind    
instruments1 and sometimes illustrated  
his remarks by showing how notes were   
varied in pitch on the old-fashioned    
,french horn1 played with the hand in   
the bell1 a performance which always    
intensely delighted the ,natural        
,philosophy ,class4  ,at the ,jubilee   
commemoration of the society1 #ahic1    
,lord ,kelvin recalled that             
,mendelssohn1 ,weber and ,beethoven were
 the gods of the infant association4    
,those of his pupils who came more      
intimately in contact with him will     
remember his keen admiration for these  
and other great composers1 especially   
,bach1 ,mozart1 and ,beethoven1 and his 
delight in hearing their works4 ,the    
,waldstein sonata was a special         
favourite4 ,it has been remarked before 
now1 and it seems to be true1 that the  

music of ,bach and ,beethoven has had   
special attractions for many great      
mathematicians4  ,at ,cambridge ,thomson
 made the acquaintance of ,george       
,gabriel ,stokes1 who graduated as      
,senior ,wrangler and ,first ,smith's   
,prizeman in #ahda1 and eight years     
later became ,lucasian ,professor of    
,mathematics in the ,university of      
,cambridge4 ,their acquaintance soon    
ripened into a close friendship1 which  
lasted until the death of ,stokes in    
#aijc4 ,the ,senior ,wrangler and the   
,peterhouse ,undergraduate undertook the
 composition of a series of notes and   
papers on points in pure and physical   
mathematics which required clearing up1 
or putting in a new point of view2 and  
so began a life-long intercourse and    
correspondence which was of great value 
to science4  ,thomson's papers of this  
period are on a considerable variety of 
subjects1 including his favourite       
subject of the flux of heat4 ,there are 
sixteen in all that seem to have been   

written and published during his        
undergraduate residence at ,cambridge4  
,most of them appeared in the ,cambridge
 ,mathematical ,journal between #ahdb   
and #ahde2 but three appeared in #ahde  
in ,liouville's ,journal de             
,mathmatiques4 ,four are on subjects of 
pure mathematics1 such as ,dupin's      
theorem regarding lines of curvature of 
orthogonally intersecting surfaces1 the 
reduction of the general equation of    
surfaces of the second order 7now called
 second degree71 six are on various     
subjects of the theory of heat1 one is  
on attractions1 five are on electrical  
theory1 and one is on the law of gravity
 at the surface of a revolving          
homogeneous fluid4 ,it is impossible to 
give an account of all these papers     
here4 ,some of them are new             
presentations or new proofs of known    
theorems1 one or two are fresh and clear
 statements of fundamental principles to
 be used later as the foundation of more
 complete statements of mathematical    

theory2 but all are marked by clearness 
and vigour of treatment4  ,another      
paper1 published in the form of a       
letter1 of date ,october #h1 #ahde1 to  
,m4 ,liouville1 and published in the    
,journal de ,mathmatiques in the same   
year1 indicates that either before or   
shortly after taking his degree1        
,thomson had invented his celebrated    
method of ,electric ,images for the     
solution of problems of electric        
distribution4 ,of this method1 which is 
one of the most elegant in the whole    
range of physical mathematics1 and      
solves at a stroke some problems1       
otherwise almost intractable1 we shall  
give some account in the following      
chapter4  ,this record of work is       
prodigious for a student reading for the
 mathematical tripos2 and it is somewhat
 of an irony of fate that such          
scientific activity is1 on the whole1   
rather a hindrance than a help in the   
preparation for that elaborate ordeal of
 examination4 ,great expectations had   

been formed regarding ,thomson's        
performance2 hardly ever before had a   
candidate appeared who had done so much 
and so brilliant original work1 and     
there was little doubt that he would be 
easily first in any contest involving   
real mathematical power1 that is1       
ability to deal with new problems and to
 express new relations of facts in      
mathematical language4 ,but the tripos  
was not a test of power merely2 it was a
 test also of acquisition1 and1 to      
candidates fairly equal in this respect1
 also of memory and of quickness of     
reproduction on paper of acquired       
knowledge4  ,the moderators on the      
occasion were ,robert ,leslie ,ellis and
 ,harvey ,goodwin1 both distinguished   
men4 ,ellis had been ,senior ,wrangler  
and first ,smith's ,prizeman a few years
 before1 and was a mathematician of     
original power and promise1 who had     
already written memoirs of great merit4 
,goodwin had been ,second ,wrangler when
 ,ellis was ,senior1 and became known to

 a later generation as ,bishop of       
,carlisle4 ,in a life of ,ellis prefixed
 to a volume of his collected papers1   
,goodwin says3--,it was in this year    
that ,professor ,w4 ,thomson took his   
degree2 great expectations had been     
excited concerning him1 and ,i remember 
,ellis remarking to me1 with a smile1   
',you and ,i are just about fit to mend 
his pens4' ,surely never was higher     
tribute paid to candidate by examiner6  
,another story1 which1 however1 does not
 seem capable of such complete          
authentication1 is told of the same     
examination1 or it may be of the        
,smith's ,prize ,examination which      
followed4 ,a certain problem was solved1
 so it is said1 in practically identical
 terms by both the ,first and ,second   
,wranglers4 ,the examiners remarked the 
coincidence1 and were curious as to its 
origin4 ,on being asked regarding it1   
the ,senior ,wrangler replied that he   
had seen the solution he gave in a paper
 which had appeared in a recent number  

of the ,cambridge ,mathematical         
,journal2 ,thomson's answer was that he 
was the author of the paper in question6
 ,thomson was ,second ,wrangler1 and    
,parkinson1 of ,st4 ,john's ,college1   
afterwards4 ,dr4 ,parkinson1 tutor of   
,st4 ,john's and author of various      
mathematical text-books1 was ,senior4   
,these positions were reversed in the   
examination for ,smith's ,prizes1 which 
was very generally regarded as a better 
test of original ability than the       
tripos1 so that the temporary           
disappointment of ,thomson's friends was
 quickly forgotten in this higher       
success4  ,the ,tripos ,examination was 
held in the early part of ,january4 ,on 
the #beth of that month ,thomson met his
 private tutor ,hopkins in the ,senior  
,wranglers' ,walk at ,cambridge1 and in 
the course of conversation referred to  
his desire to obtain a copy of ,green's 
',essay' 7supra1 p4 #ba74 ,hopkins at   
once took him to the rooms where he had 
attended almost daily for a considerable

 time as a pupil1 and produced no less  
than three copies of the ,essay1 and    
gave him one of them4 ,a hasty perusal  
showed ,thomson that all the general    
theorems of attractions contained in his
 paper ,on the ,uniform ,motion of      
,heat1 etc41 as well as those of ,gauss 
and ,chasles1 had been set forth by     
,green and were derivable from a general
 theorem of analysis whereby a certain  
integral taken throughout a space       
bounded by surfaces fulfilling a certain
 condition is expressed as two          
integrals1 one taken throughout the     
space1 the other taken over the bounding
 surface or surfaces4  ,it has been     
stated in the last chapter that ,thomson
 had established1 as a deduction from   
the flow of heat in a uniform solid from
 sources distributed within it1 the     
remarkable theorem of the replacement1  
without alteration of the external flow1
 of these sources by a certain          
distribution over any surface of uniform
 temperature1 and had pointed out the   

analogue of this theorem in electricity4
 ,this method of proof was perfectly    
original and had not been anticipated1  
though the theorem1 as has been stated1 
had already been given by ,green and by 
,gauss4 ,in the paper entitled          
,propositions in the ,theory of         
,attraction1 published in the ,cambridge
 ,mathematical ,journal in ,november    
#ahdb1 ,thomson gave an analytical proof
 of this great theorem1 but afterwards  
found that this had been done almost    
contemporaneously by ,sturm in          
,liouville's ,journal4  ,soon after the 
,tripos and ,smith's ,prize             
,examinations were over1 ,thomson went  
to ,london1 and visited ,faraday in his 
laboratory in the ,royal ,institution4  
,then he went on to ,paris with his     
friend ,hugh ,blackburn1 and spent the  
summer working in ,regnault's famous    
laboratory1 making the acquaintance of  
,liouville1 ,sturm1 ,chasles1 and other 
,french mathematicians of the time1 and 
attending meetings of the ,acadmie des  

,sciences4 ,he made known to the        
mathematicians of ,paris ,green's       
',essay1' and the treasures it          
contained1 and frequently told in after 
years with what astonishment its results
 were received4 ,he used to relate that 
one day1 while he and ,blackburn sat in 
their rooms1 they heard some one come   
panting up the stair4 ,sturm burst in   
upon them in great excitement1 and      
exclaimed1 ,vous avez un ,mmoire de     
,green6 ,m4 ,liouville me l'a dit4 ,he  
sat down and turned over the pages of   
the ',essay1' looking at one result     
after another1 until he came to a       
complete anticipation of his proof of   
the replacement theorem4 ,he jumped up1 
pointed to the page1 and cried out1     
,voila mon affaire6  ,to this visit to  
,paris ,thomson often referred in later 
life with grateful recognition of       
,regnault's kindness1 and admiration of 
his wonderful experimental skill4 ,the  
great experimentalist was then engaged  
in his researches on the thermal        

constants of bodies1 with the elaborate 
apparatus which he designed for himself1
 and with which he was supplied by the  
wise liberality of the ,french          
,government4 ,this initiation into      
laboratory work bore fruit not long     
after in the establishment of the       
,glasgow ,physical ,laboratory1 the     
first physical laboratory for students  
in this country4  ,it is a striking     
testimony to ,thomson's genius that1 at 
the age of only seventeen1 he had       
arrived at such a fundamental and       
general theorem of attractions1 and had 
pointed out its applications to         
electrical theory4 ,and it is also very 
remarkable that the theorem should have 
been proved within an interval of two or
 three years by three different authors1
 two of them--,sturm and ,gauss--already
 famous as mathematicians4 ,green's     
treatment of the subject was1 however1  
the most general and far-reaching1 for1 
as has been stated1 the theorem of      
,gauss1 ,sturm1 and ,thomson was merely 

a particular case of a general theorem  
of analysis contained in ,green's       
',essay4' ,it has been said in jest1 but
 not without truth1 that physical       
mathematics is made up of continued     
applications of ,green's theorem4 ,of   
this enormously powerful relation1 a    
more lately discovered result1 which is 
very fundamental in the theory of       
functions of a complex variable1 and    
which is generally quoted as ,riemann's 
theorem1 is only a particular case4     
,thomson had the greatest reverence for 
the genius of ,green1 and found in his  
memoirs1 and in those of ,cauchy on wave
 propagation1 the inspiration for much  
of his own later work4#g ,in #ahej he   
obtained the republication of ,green's  
',essay' in ,crelle's ,journal2 in later
 years he frequently expressed regret   
that it had not been published in       
,england4  ,in the commencement of #ahde
 ,thomson told ,liouville of the method 
of ,electric ,images which he had       
discovered for the solution of problems 

of electric distribution4 ,on ,october  
#h1 #ahde1 after his return to          
,cambridge1 he wrote to ,liouville a    
short account of the results of the     
method in a number of different cases1  
and in two letters written on ,june #bf 
and ,september #af of the following     
year1 he stated some further results1   
including the solution of the problem of
 the distribution upon a spherical bowl 
7a segment of a spherical conducting    
shell made by a plane section7 insulated
 and electrified4 ,this last very       
remarkable result was given without     
proof1 and remained unproved until      
,thomson published his demonstration    
twenty-three years later in the         
,philosophical ,magazine4#h ,this had   
been preceded by a series of papers in  
,march1 ,may1 and ,november #ahdh1      
,november #ahdi1 and ,february #ahej1 in
 the ,cambridge and ,dublin             
,mathematical ,journal1 on various parts
 of the mathematical theory of          
electricity in equilibrium1#i in which  

the theory of images is dealt with4 ,the
 letters to ,liouville promptly appeared
 in the ,journal1 and the veteran       
analyst wrote a long ,note on their     
subject1 which concludes as follows3    
,mon but sera rempli1 je le rpte1 s'ils 
ces dveloppements peuvent aider  bien   
faire comprendre la haute importance du 
travail de ce jeune gomtre1 et si ,m4   
,thomson lui-mme veut bien y voir une   
preuve nouvelle de l'amiti que je lui   
porte et de l'estime qui j'ai pour son  
talent4  ,the method of images may be   
regarded as a development in a          
particular direction of the paper ,on   
the ,uniform ,motion of ,heat already   
referred to1 and1 taken along with this 
latter paper1 forms the most striking   
indication afforded by the whole range  
of ,thomson's earlier work of the       
strength and originality of his         
mathematical genius4 ,accordingly a     
chapter is here devoted to a more       
complete explanation of the first paper 
and the developments which flowed from  

it4 ,the general reader may pass over   
the chapter1 and return to it from time 
to time as he finds opportunity1 until  
it is completely understood4            
,,chapter ,,iv  ,,the ,,mathematical    
,,theory ,,of ,,electricity ,,in        
,,equilibrium4 ,,electric ,,images4     
,,electric ,,inversion   ,in describing 
,thomson's early electrical researches  
we shall not enter into detailed        
calculations1 but merely explain the    
methods employed4 ,the meaning of       
certain technical terms may be recalled 
in the first place4  ,the whole space in
 which a distribution of electricity    
produces any action on electrified      
bodies is called the electrical field of
 the distribution4 ,the force exerted on
 a very small insulated trial conductor1
 on which is an electric charge of      
amount equal to that taken as the unit  
quantity of electricity1 measures the   
field-intensity at any point at which   
the conductor is placed4 ,the direction 
of the field-intensity at the point is  

that in which the small conductor is    
there urged4 ,if the charge on the small
 conductor were a negative unit1 instead
 of a positive1 the direction of the    
force would be reversed2 the magnitude  
of the force would remain the same4 ,to 
make the field-intensity quite definite1
 a positive unit is chosen for its      
specification4 ,for a charge on the     
trial-conductor consisting of any number
 of units1 the force is that number of  
times the field-intensity4 ,the         
field-intensity is often specified by   
its components1 ,x1 ,y1 ,z in three     
chosen directions at right angles to one
 another4  ,now in all cases in which   
the action1 whether attraction or       
repulsion1 between two unit quantities  
of matter concentrated at points is     
inversely as the square of the distance 
between the charges1 the                
field-intensity1 or its components1 can 
be found from a certain function ,v of  
the charges forming the acting          
distribution which is always capable of 

being regarded for mathematical purposes
 as a system of small charges existing  
at points of space1 point-charges we    
shall call them1 their positions1 and   
the position of the point at which the  
field-intensity is to be found4 ,if q1  
q1 444 be the point-charges1 and be     
positive when the charges are positive  
and negative when the charges are       
negative1 and r1 r1 444 be their        
distances from the point ,p1 ,v is qr   
qr  444 ,the field-intensity is the rate
 of diminution of the value of ,v at ,p1
 taken along the specified direction4   
,the three gradients parallel to the    
three chosen coordinate directions are  
,x1 ,y1 ,z2 but for their calculation it
 is necessary to insert the values of r1
 r1 444 in terms of the coordinates     
which specify the positions of the      
point-charges1 and the coordinates x1 y1
 z which specify the position of ,p4    
,once this is done1 ,x1 ,y1 ,z are      
obtained by a simple systematic process 
of calculation1 namely1 differentiation 

of the function ,v with respect to x1 y1
 z4  ,this function ,v seems to have    
been first used by ,laplace for         
gravitational matter in the ,mcanique   
,cleste2 its importance for electricity 
and magnetism was recognised by ,green1 
who named it the potential4 ,it has an  
important physical signification4 ,it   
represents the work which would have to 
be done to bring a unit of positive     
electricity1 against the electrical     
repulsion of the distribution1 up to the
 point ,p from a point at an infinite   
distance from every part of the         
distribution2 or1 in other words1 what  
we now call the potential energy of a   
charge q situated at ,p is q,v4 ,the    
excess of the potential at ,p1 over the 
potential at any other point ,q in the  
field1 is the work which must be spent  
in carrying a positive unit from ,q to  
,p against electrical repulsion4 ,of    
course1 if the force to be overcome from
 ,q to ,p is on the whole an attraction1
 work has not been spent in effecting   

the transference1 but gained by allowing
 it to take place4 ,the difference of   
potential is then negative1 that is1 the
 potential of ,q is higher than that of 
,p4  ,the difference of potential       
depends only on the points ,p and ,q1   
and not at all on the path pursued      
between them4 ,thus1 if a unit of       
electricity be carried from ,p to ,q by 
any path1 and back by any other1 no work
 is done on the whole by the agent      
carrying the unit4 ,this simple fact    
precludes the possibility of obtaining a
 so-called perpetual motion 7a          
self-acting machine doing useful work7  
by means of electrical action4 ,the same
 thing is true mutatis mutandis of      
gravitational action4  ,in the thermal  
analogy explained by ,thomson in his    
first paper1 the positive point-charges 
are point-sources of heat1 which is     
there poured at constant rate into the  
medium 7supposed of uniform quality7 to 
be drawn off in part from the medium at 
constant rate where there are sinks 7or 

negative sources71--the negative        
point-charges in the electrical         
case1--while the remainder is conducted 
away to more and more distant parts of  
the conducting medium supposed          
infinitely extended4 ,whenever a        
point-source1 or a point-sink1 exists at
 a distance from other sources or sinks1
 the flow in the vicinity is in straight
 lines from or to the point1 and these  
straight lines would be indefinitely    
extended if either source or sink       
existed by itself4 ,as it is1 the       
direction and amount of flow everywhere 
depends on the flow resulting from the  
whole arrangement of sources and sinks4 
,lines can be drawn in the medium which 
show the direction of the resultant flow
 from point to point1 and these lines of
 flow can be so spaced as to indicate1  
by their closeness together or their    
distance apart1 where the rate of flow  
is greater or smaller2 and such lines   
start from sources1 and either end in   
sinks or continue their course to       

infinity4 ,in the electrical case these 
lines are the analogues of the lines of 
electric force 7or field-intensity7 in  
the insulating medium1 which start from 
positive charges and end in negative1 or
 are prolonged to infinity4  ,across    
such lines of flow can be drawn a family
 of surfaces1 to each of which the lines
 met by the surface are perpendicular4  
,these surfaces are the equitemperature 
surfaces1 or1 as they are usually       
called1 the isothermal surfaces4 ,they  
can be drawn more closely crowded       
together1 or more widely separated1 so  
as to indicate where the rate of falling
 off of temperature 7the temperature    
slope7 is greater or less1 just as the  
contour lines in a map show the slopes  
on a hill-side4  ,instead of the thermal
 analogy might have been used equally   
well that of steady flow in an          
indefinitely extended mass of           
homogeneous frictionless and            
incompressible fluid1 into which fluid  
is being poured at a constant rate by   

sources and withdrawn by sinks4 ,the    
isothermal surfaces are replaced by     
surfaces of equal pressure1 while lines 
of flow in one are also lines of flow in
 the other4  ,now let heat be poured    
into the medium at constant rate by a   
single point-source ,p 7,fig4 #a71 and  
drawn off at a smaller rate by a single 
point-sink ,p'1 while the remainder     
flows to more and more remote parts of  
the medium1 supposed infinite in extent 
in every direction4 ,after a sufficient 
time from the beginning of the flow a   
definite system of lines of flow and    
isothermal surfaces can be traced for   
this case in the manner described above4
 ,one of the isothermal surfaces will be
 a sphere ,s surrounding the sink1      
which1 however1 will not be at the      
centre of the sphere1 but so situated   
that the source1 sink1 and centre are in
 line1 and that the radius of the sphere
 is a mean proportional between the     
distances of the source and sink from   
the centre4 ,if a be the radius of the  

sphere and f the distance of the source 
from the centre of the sphere1 the heat 
carried off by the sink is the fraction 
af of that given out by the source4     
,illustration3 ,,fig4 #a4  ,in the      
electrical analogue1 the source and sink
 are respectively a point-charge and    
what is called the electric image of    
that charge with respect to the sphere1 
which is in this case an equipotential  
surface4 ,and just as the lines of flow 
of heat meet the spherical isothermal   
surface at right angles1 so the lines of
 force in the electrical case meet the  
equipotential surface also at right     
angles4 ,now obviously in the thermal   
case a spherical sink could be arranged 
coinciding with the spherical surface so
 as to receive the flow there arriving  
and carry off the heat from the medium1 
without in the least disturbing the flow
 outside the sphere4 ,the whole amount  
of heat arriving would be the same3 the 
amount received per unit area at any    
point on the sphere would evidently be  

proportional to the gradient of         
temperature there towards the surface4  
,of course the same thing could be done 
at any isothermal surface1 and the same 
proportionality would hold in that case4
  ,similarly the source could be        
replaced by a surface-distribution of   
sources over any surrounding isothermal 
surface2 and the condition to be        
fulfilled in that case would be that the
 amount of heat given out per unit area 
anywhere should be exactly that which   
flows out along the lines of flow there 
in the actual case4 ,outside the surface
 the field of flow would not be affected
 by this replacement4 ,it is obvious    
that in this case the outflow per unit  
area must be proportional to the        
temperature slope outward from the      
surface4  ,the same statements hold for 
any complex system of sources and sinks4
 ,there must be the same outflow from   
the isothermal surface or inflow towards
 it1 as there is in the actual case1 and
 the proportionality to temperature     

slope must hold4  ,this is exactly      
analogous to the replacement by a       
distribution on an equipotential surface
 of the electrical charge or charges    
within the surface1 by a distribution   
over the surface1 with fulfilment of    
,coulomb's theorem 7p4 #dc below7 at the
 surface4 ,thomson's paper on the       
,uniform ,motion of ,heat gave an       
intuitive proof of this great theorem of
 electrostatics1 which the statements   
above may help to make clear to those   
who have1 or are willing to acquire1    
some elementary knowledge of            
electricity4  ,returning to the         
distribution on any isothermal surface  
surrounding the sink 7or sinks7 we see  
that it represents a surface-sink in    
equilibrium with the flow in the field4 
,the distribution on a metal shell1     
coinciding with the surface1 which keeps
 the surface at a potential which is the
 analogue of the temperature at the     
isothermal surface1 while the shell is  
under the influence of a point-charge of

 electricity--the analogue of the       
thermal source--is the distribution as  
affected by the induction of the        
point-charge4 ,if the shell coincide    
with the spherical equipotential surface
 referred to above1 and the distribution
 given by the theorem of replacement be 
made upon it1 the shell will be at zero 
potential1 and the charge will be that  
which would exist if the shell were     
uninsulated1 that is1 the induced       
charge4  ,the consideration of the      
following simple problem will serve to  
make clear the meaning of an electric   
image1 and form a suitable introduction 
to a description of the application of  
the method to the electrification of    
spherical surfaces4 ,imagine a very     
large plane sheet of tinfoil connected  
by a conducting wire with the earth4 ,if
 there are no electrified bodies near1  
the sheet will be unelectrified4 ,but   
let a very small metallic ball with a   
charge of positive electricity upon it  
be brought moderately close to one face 

of the tinfoil4 ,the tinfoil will be    
electrified negatively by induction1 and
 the distribution of the negative charge
 will depend on the position of the     
ball4 ,now1 it can be shown that the    
field of electric force1 on the same    
side of the tinfoil as the ball1 is     
precisely the same as would be produced 
if the foil 7and everything behind it7  
were removed1 and an equal negative     
charge of electricity placed behind the 
tinfoil on the prolonged perpendicular  
from the ball to the foil1 and as far   
from the foil behind as the ball is from
 it in front4 ,such a negative charge   
behind the tinfoil sheet is called an   
electric image of the positive charge in
 front4 ,it is situated1 as will be seen
 at what would be1 if the tinfoil were a
 mirror1 the optical image of the ball  
in the mirror4  ,illustration3 ,,fig4   
#b4  ,now1 suppose a second very large  
sheet of tinfoil to be placed parallel  
to the first sheet1 so that the small   
electrified sphere is between the two   

sheets1 and that this second sheet is   
also connected to the earth4 ,the charge
 on the ball induces negative           
electricity on both sheets1 but besides 
this each sheet by its charge influences
 the other4 ,the problem of distribution
 is much more complicated than in the   
case of a single sheet1 but its solution
 is capable of very simple statement4   
,let us call the two sheets ,a and ,b   
7,fig4 #b71 and regard them for the     
moment as mirrors4 ,a first image of an 
object ,p between the two mirrors is    
produced directly by each1 but the image
 ,i in ,a is virtually an object in     
front of ,b1 and the image ,j in ,b an  
object in front of ,a1 so that a second 
image more remote from the mirror than  
the first is produced in each case4     
,these second images ,i and ,j in the   
same way produce third images still more
 remote1 and so on4 ,the positions are  
determined just as for an object and a  
single mirror4 ,there is thus an        
infinite trail of images behind each    

mirror1 the places of which any one can 
assign4  ,illustration3 ,,fig4 #c4      
,every one may see the realisation of   
this arrangement in a shop window1 the  
two sides of which are covered by       
parallel sheets of mirror-glass4 ,an    
infinite succession of the objects in   
the window is apparently seen on both   
sides4 ,when the objects displayed are  
glittering new bicycles in a row the    
effect is very striking2 but what we are
 concerned with here is a single small  
object like the little ball1 and its two
 trails of images4 ,the electric force  
at any point between the two sheets of  
tinfoil is exactly the same as if the   
sheets were removed and charges         
alternately negative and positive were  
placed at the image-points1 negative at 
the first images1 positive at the second
 images1 and so on1 each charge being   
the same in amount as that on the ball4 
,we have an electric kaleidoscope with  
parallel mirrors4 ,when the angle       
between the conducting planes is an     

aliquot part of #cfj1 let us say #fj1   
the electrified point and the images are
 situated1 just as are the object and   
its image in ,brewster's kaleidoscope1  
namely at the angular points of a       
hexagon1 the sides of which are         
alternately 7as shown in ,fig4 #c7 of   
lengths twice the distance of the       
electrified point from ,a and from ,b4  
,illustration3 ,,fig4 #d4  ,now consider
 the spherical surface referred to at p4
 #cg1 which is kept at uniform potential
 by a charge at the external point ,p1  
and a charge q' at the inverse point ,p'
 within the sphere4 ,if ,e 7,fig4 #d7 be
 any point whatever on the surface1 and 
r1 r' be its distances from ,p and ,p'1 
it is easy to prove by geometry that the
 two triangles ,,cpe and ,,cep' are     
similar1 and therefore r'  raf4 ,here af
 is used to mean a divided by f4 ,the   
mark  is adopted instead of the usual   
bar of the fraction1 for convenience of 
printing4 ,now1 by the explanation given
 above1 the potential produced at any   

point by a charge q at another point1 is
 equal to the ratio of the charge q to  
the distance between the points4 ,thus  
the potential at ,e due to the charge q 
at ,p is qr1 and that at ,e due to a    
charge q' at ,p' is q'r'4 ,thus if q'   
qaf1 q' at ,p' will produce a potential 
at ,e  qafr'  qr1 by the value of r4    
,hence q at ,p and -qaf at ,p'          
coexisting will give potential qr  qr or
 zero1 at ,e4 ,thus the charge qaf1 at  
the internal point ,p' will in presence 
of q at ,p keep all points of the       
spherical surface at zero potential4    
,these two charges represent the source 
and sink in the thermal analogue of p4  
#cg above4  ,now replace ,s by a        
spherical shell of metal connected to   
the earth by a long fine wire1 and      
imagine all other conductors to be at a 
great distance from it4 ,if this be     
under the influence of the charge q at  
,p alone1 a charge is induced upon it   
which1 in presence of ,p1 maintains it  
at zero potential4 ,the internal charge 

qaf1 and the induced distribution on the
 shell are thus equivalent as regards   
the potential produced by either at the 
spherical surface2 for each counteracts 
then the potential produced by q at ,p4 
,but it can be proved that if a         
distribution over an equipotential      
surface can be made to produce the same 
potential over that surface as a given  
internal distribution does1 they produce
 the same potentials at all external    
points1 or1 as it is usually put1 the   
external fields are the same4 ,this is  
part of the statement of what has been  
called the theorem of replacement       
discovered by ,green1 ,gauss1 ,thomson1 
and ,chasles as described above4        
,another part of the statement of the   
theorem may now be formulated4 ,coulomb 
showed long ago that the surface-density
 of electricity at any point on a       
conductor is proportional to the        
resultant field-intensity just outside  
the surface at that point4 ,since the   
surface is throughout at one potential  

this intensity is normal to the surface4
 ,let it be denoted by ,n1 and s be the 
surface-density3 then according to the  
system of units usually adopted #ds  ,n4
  ,let now the rate of diminution of    
potential per unit of distance outwards 
7or downward gradient of potential7 from
 the equipotential surface be determined
 for every point of the surface1 and let
 electricity be distributed over the    
surface1 so that the amount per unit    
area at each point 7the surface-density7
 is made numerically equal to the       
gradient there divided by #d4 ,this1 by 
,coulomb's law1 stated above1 gives that
 field-intensity just outside the       
surface which exists for the actual     
distribution1 and therefore1 as can be  
proved1 gives the same field everywhere 
else outside the surface4 ,the external 
fields will therefore be equivalent1 and
 further1 the amount of electricity on  
the surface will be the same as that    
situated within it in the actual        
distribution4  ,thus it is only         

necessary to find for qaf at ,p' and q  
at ,p1 the falling off gradient ,n of   
potential outside the spherical surface 
at any point ,e1 and to take ,n#d1 to   
obtain s the surface-density at ,e4     
,calculation of this gradient for the   
sphere gives #ds  q7f  a7ar4 ,the       
surface-density is thus inversely as the
 cube of the distance ,,pe4  ,if the    
influencing point ,p be situated within 
the spherical shell1 and the shell be   
connected to earth as before1 the       
induced distribution will be on its     
interior surface4 ,the corresponding    
point ,p will now be outside1 but given 
by the same relation4 ,and a will now be
 greater than f1 and the density will be
 given by #ds  q7a  f7ar1 where1 f and r
 have the same meanings with regard to  
,e and ,p as before4  ,p' is in each    
case called the image of ,p in the      
sphere ,s1 and the charge qaf there     
supposed situated is the electric image 
of the charge q at ,p4 ,it will be seen 
that an electric image is a charge1 or  

system of charges1 on one side of an    
electrified surface which produces on   
the other side of that surface the same 
electrical field as is produced by the  
actual electrification of the surface4  
,while by the theorem of replacement    
there is only one distribution over a   
surface which produces at all points on 
one side of a surface the same field as 
does a distribution ,d on the other side
 of the surface1 this surface           
distribution may be equivalent to       
several different arrangements of ,d4   
,thus the point-charge at ,p' is only   
one of various image-distributions      
equivalent to the surface-distribution  
in the sense explained4 ,for example1 a 
uniform distribution over any spherical 
surface with centre at ,p' 7,fig4 #d7   
would do as well1 provided this         
spherical surface were not large enough 
to extend beyond the surface ,s4  ,in   
order to find the potential of the      
sphere 7,fig4 #d7 when insulated with a 
charge ,q upon it1 in presence of the   

influencing charge q at the external    
point ,p1 it is only necessary to       
imagine uniformly distributed over the  
sphere1 already electrified in the      
manner just explained1 the charge ,q    
aqf4 ,then the whole charge will be ,q1 
and the uniformity of distribution will 
be disturbed1 as required by the action 
of the influencing point-charge4 ,the   
potential will be ,qa  qf4 ,for a given 
potential ,v of the sphere1 the total   
charge is a,v  aqf1 that is the charge  
is a,v over and above the induced       
charge4  ,if instead of a single        
influencing point-charge at ,p there be 
a system of influencing point-charges at
 different external points1 each of     
these has an image-charge to be found in
 amount and situation by the method just
 described1 and the induced distribution
 is that obtained by superimposing all  
the surface distributions found for the 
different influencing points4  ,the     
force of repulsion between the          
point-charge q and the sphere 7with     

total charge ,q7 can be found at once by
 calculating the sum of the forces      
between q at ,p and the charges ,q  aqf 
at ,c and aqf at ,p'4  ,this can be     
found also by calculating the energy of 
the system1 which will be found to      
consist of three terms1 one representing
 the energy of the sphere with charge ,q
 uninfluenced by an external charge1 one
 representing the energy on a small     
conductor 7not a point7 at ,p existing  
alone1 and a third representing the     
mutual energy of the electrification on 
the sphere and the charge q at ,p       
existing in presence of one another4 ,by
 a known theorem the energy of a system 
of conductors is one half of the sum    
obtained by multiplying the potential of
 each conductor by its charge and adding
 the products together4 ,it is only     
necessary then to find the variation of 
the last term caused by increasing f by 
a small amount df4 ,this will be the    
product ,f4df of the force ,f required  
and the displacement4  ,either method   

may be applied to find the forces of    
attraction and repulsion for the systems
 of electrified spheres described below4
  ,the problem of two mutually          
influencing non-intersecting spheres1   
,s1 ,s 7,fig4 #e71 insulated with given 
charges1 q1 q1 may now be dealt with in 
the following manner4 ,let each be      
supposed at first charged uniformly4 ,by
 the known theorem referred to above1   
the external field of each is the same  
as if its whole charge were situated at 
the centre4 ,now if the distribution on 
,s1 say1 be kept unaltered1 while that  
on ,s is allowed to change1 the action  
of ,s on ,s is the same as if the charge
 q were at the centre ,c of ,s4 ,thus if
 f be the distance between the centres  
,c1 ,c1 and a be the radius of ,s1 the  
distribution will be that corresponding 
to q  aqf uniformly distributed on ,s   
together with the induced charge aqf1   
which corresponds to the image-charge at
 the point ,i 7within ,s71 the inverse  
of ,c with respect to ,s4 ,now let the  

charge on ,s be fixed in the state just 
supposed while that on ,s is freed4 ,the
 charge on ,s will rearrange itself     
under the influence of q  aqf 7  q'7 and
 aqf1 considered as at ,c and ,i        
respectively4 ,the former of these will 
give a distribution equivalent to q     
aq'f uniformly distributed over ,s1 and 
an induced distribution of amount aq'f  
at ,j1 the inverse point of ,c with     
regard to ,s4 ,the image-charge aqf at  
,i in ,s will react on ,s and give an   
induced distribution a7aqf7f'1 7,i,c    
f'7 corresponding to an image-charge    
aaqff' at the inverse point ,j of ,p    
with respect to ,c,s4 ,thus the         
distribution on ,s is equivalent to q   
aq'f  aaqff' distributed uniformly over 
it1 together with the two induced       
distributions just described4           
,illustration3 ,,fig4 #e4  ,in the same 
way these two induced distributions on  
,s may now be regarded as reacting on   
the distribution on ,s as would         
point-charges aqf and aaqff'1 situated  

at ,j and ,j respectively1 and would    
give two induced distributions on ,s    
corresponding to their images in ,s4    
,thus by partial influences in unending 
succession the equilibrium state of the 
two spheres could be approximated to as 
nearly as may be desired4 ,an infinite  
trail of electric images within each of 
the two spheres is thus obtained1 and   
the final state of each conductor can be
 calculated by summation of the effects 
of each set of images4  ,if the final   
potentials1 ,v1 ,v1 say1 of the spheres 
are given the process is somewhat       
simpler4 ,let first the charges be      
supposed to exist uniformly distributed 
over each sphere1 and to be of amount   
a,v1 a,v in the two cases4 ,the uniform 
distribution on ,s will raise the       
potential of ,s above ,v1 and to bring  
the potential down to ,v in presence of 
this distribution we must place an      
induced distribution over ,s1           
represented as regards the external     
field by the image-charge aa,vf 7at the 

image of ,c in ,s7 where f is the       
distance between the centres4 ,the      
charge a,v on ,s will similarly have an 
action on ,s to be compensated in the   
same way by an image-charge aa,vf at the
 image of ,c in ,s4 ,now these two      
image-charges will react on the spheres 
,s and ,s respectively1 and will have to
 be balanced by induced distributions   
represented by second image-charges1 to 
be found in the manner just exemplified4
 ,these will again react on the spheres 
and will have to be compensated as      
before1 and so on indefinitely4 ,the    
charges diminish in amount1 and their   
positions approximate more and more1    
according to definite laws1 and the     
final state is to be found by summation 
as before4  ,the force of repulsion is  
to be found by summing the forces       
between all the different pairs of      
charges which can be formed by taking   
one charge of each system at its proper 
point3 or it can be obtained by         
calculating the energy of the system4   

,the method of successive influences was
 given originally by ,murphy1 but the   
mode of representing the effects of the 
successive induced charges by           
image-charges is due to ,thomson4 ,quite
 another solution of this problem is1   
however1 possible by ,thomson's method  
of electrical inversion4  ,a similar    
process to that just explained for two  
charged and mutually influencing spheres
 will give the distribution on two      
concentric conducting spheres1 under the
 influence of a point-charge q at ,p    
between the inner surface of the outer  
and the outer surface of the inner1 as  
shown in ,fig4 #g4 ,there the influence 
of q at ,p1 and of the induced          
distributions on one another1 is        
represented by two series of images1 one
 within the inner sphere and one outside
 the outer4 ,these charges and positions
 can be calculated from the result for a
 single sphere and point-charge4        
,thomson's method of electrical         
inversion1 referred to above1 enabled   

the solutions of unsolved problems to be
 inferred from known solutions of       
simpler cases of distribution4 ,we give 
here a brief account of the method1 and 
some of its results4 ,first we have to  
recall the meaning of geometrical       
inversion4 ,in ,fig4 #f the distances   
,,op1 ,,op1 ,,oq1 ,,oq' fulfil the      
relation ,,op4,,op'  ,,oq4,,oq'  a4     
,thus ,p' is 7see p4 #cg7 the inverse of
 the point ,p with respect to a sphere  
of radius a and centre ,o 7indicated by 
the dotted line in ,fig4 #f71 and       
similarly ,q' is the inverse of ,q with 
respect to the same sphere and centre4  
,o is called the centre of inversion1   
and the sphere of radius a is called the
 sphere of inversion4 ,thus the sphere  
of ,figs4 #a and #d is the sphere of    
inversion for the points ,p and ,p'1    
which are inverse points of one another4
 ,for any system of points ,p1 ,q1 4441 
another system ,p'1 ,q'1 444 of inverse 
points can be found1 and if the first   
system form a definite locus1 the second

 will form a derived locus1 which is    
called the inverse of the former4 ,also 
if ,p'1 ,q'1 444 be regarded as the     
direct system1 ,p1 ,q1 444 will be the  
corresponding inverse system with regard
 to the same sphere and centre4 ,p' is  
the image of ,p1 and ,p is the image of 
,p'1 and so on1 with regard to the same 
sphere and centre of inversion4         
,illustration3 ,,fig4 #f4  ,the inverse 
of a circle is another circle1 and      
therefore the inverse of a sphere is    
another sphere1 and the inverse of a    
straight line is a circle passing       
through the centre of inversion1 and of 
an infinite plane a sphere passing      
through the centre of inversion4        
,obviously the inverse of a sphere      
concentric with the sphere of inversion 
is a concentric sphere4  ,the line      
,p',q' is of course not the inverse of  
the line ,,pq1 which has for its inverse
 the circle passing through the three   
points ,o1 ,p'1 ,q'1 as indicated in    
,fig4 #f4  ,the following results are   

easily proved4  ,a locus and its inverse
 cut any line ,,op at the same angle4   
,to a system of point-charges q1 q1 444 
at points ,p1 ,p1 444 on one side of the
 surface of the sphere of inversion     
there is a system of charges aqf1 aqf1  
444 on the other side of the spherical  
surface ,,op  f1 ,,op  f4 ,this inverse 
system1 as we shall call it1 produces   
the same potential at any point of the  
sphere of inversion1 as does the direct 
system from which it is derived4  ,if   
,v1 ,v' be the potentials produced by   
the whole direct system at ,q1 and by   
the whole inverse system at ,q'1 ,v',v  
ra  ar'1 where ,,oq  r1 ,,oq'  r'4      
,thus if ,v is constant over any surface
 ,s'1 ,v' is not a constant over the    
inverse surface ,s'1 unless r is a      
constant1 that is1 unless the surface   
,s' is a sphere concentric with the     
sphere of inversion1 in which case the  
inverse surface is concentric with it   
and is an equipotential surface of the  
inverse distribution4  ,further1 if q be

 distributed over an element d,s of a   
surface1 the inverse charge aqf will be 
distributed over the corresponding      
element d,s' of the inverse surface4    
,but d,s'd,s  af  f'a where f1 f' are   
the distances of ,o from d,s and d,s'4  
,thus if s be the density on d,s and s' 
the inverse density on d,s' we have s's 
af'  fa4  ,when ,v is constant over the 
direct surface1 while r has different   
values for different directions of ,,oq1
 the different points of the inverse    
surface may be brought to zero potential
 by placing at ,o a charge a,v4 ,for    
this will produce at ,q' a potential    
a,vr' which with ,v' will give at ,q' a 
potential zero4 ,this shows that ,v' is 
the potential of the induced            
distribution on ,s' due to a charge a,v 
at ,o1 or that ,v' is the potential due 
to the induced charge on ,s' produced by
 the charge a,v at ,o4  ,thus we have   
the conclusion that by the process of   
inversion we get from a distribution in 
equilibrium1 on a conductor of any form1

 an induced distribution on the inverse 
surface supposed insulated and          
conducting2 and conversely we obtain    
from a given induced distribution on an 
insulated conducting surface1 a natural 
equilibrium distribution on the inverse 
surface4 ,in each case the inducing     
charge is situated at the centre of     
inversion4 ,the charges on the conductor
 7or conductors7 after inversion are    
always obtainable at once from the fact 
that they are the inverses of the       
charges on the conductor 7or conductors7
 in the direct case1 and the            
surface-densities or volume-densities   
can be found from the relations stated  
above4  ,illustration3 ,,fig4 #g4  ,now 
take the case of two concentric spheres 
insulated and influenced by a           
point-charge q placed at a point ,p     
between them as shown in ,fig4 #g4 ,we  
have seen at p4 #di how the induced     
distribution1 and the amount of the     
charge1 on each sphere is obtained from 
the two convergent series of images1 one

 outside the outer sphere1 the other    
inside the inner sphere4 ,we do not here
 calculate the density of distribution  
at any point1 as our object is only to  
explain the method2 but the quantities  
on the spheres ,s and ,s1 are           
respectively q4,,oa4,,pb7,,op4,,ab71    
q4,,ob4,,ap7,,op4,,ab74  ,it may be     
noticed that the sum of the induced     
charges is q1 and that as the radii of  
the spheres are both made indefinitely  
great1 while the distance ,,ab is kept  
finite1 the ratios ,,oa,,op1 ,,ob,,op   
approximate to unity1 and the charges to
 q4,,pb,,ab1 q4,,ap,,ab1 that is1 the   
charges are inversely as the distances  
of ,p from the nearest points of the two
 surfaces4 ,but when the radii are made 
indefinitely great we have the case of  
two infinite plane conducting surfaces  
with a point-charge between them1 which 
we have described above4  ,now let this 
induced distribution1 on the two        
concentric spheres1 be inverted from ,p 
as centre of inversion4 ,we obtain two  

non-intersecting spheres1 as in ,fig4   
#e1 for the inverse geometrical system1 
and for the inverse electrical system an
 equilibrium distribution on these two  
spheres in presence of one another1 and 
charged with the charges which are the  
inverses of the induced charges4 ,these 
maintain the system of two spheres at   
one potential4 ,from this inversion it  
is possible to proceed as shown by      
,maxwell in his ,electricity and        
,magnetism1 vol4 i1  #agc1 to the       
distribution on two spheres at two      
different potentials2 but we have shown 
above how the problem may be dealt with 
directly by the method of images4       
,illustration3 ,,fig4 #h4  ,again take  
the case of two parallel infinite planes
 under the influence of a point-charge  
between them4 ,this system inverted from
 ,p as centre gives the equilibrium     
distribution on two charged insulated   
spheres in contact 7,fig4 #h72 for this 
system is the inverse of the planes and 
the charges upon them4 ,another         

interesting case is that of the electric
 kaleidoscope referred to above4 ,here  
the two infinite conducting planes are  
inclined at an angle #cfjn1 where n is a
 whole number1 and are therefore bounded
 in one direction by the straight line  
which is their intersection4 ,the image 
points ,i1 ,j1 4441 of ,p placed in the 
angle between the planes are situated as
 shown in ,fig4 #c1 and are n  #a in    
number4 ,this system inverted from ,p as
 centre gives two spherical surfaces    
which cut one another at the same angle 
as do the planes4 ,this system is one of
 electrical equilibrium in free space1  
and therefore the problem of the        
distribution on two intersecting spheres
 is solved1 for the case at least in    
which the angle of intersection is an   
aliquot part of #cfj4 ,when the planes  
are at right angles the result is that  
for two perpendicularly intersecting    
planes1 for which ,fig4 #i gives a      
diagram4  ,illustration3 ,,fig4 #i4     
,but the greatest achievement of the    

method was the determination of the     
distribution on a segment of a thin     
spherical shell with edge in one plane4 
,the solution of this problem was       
communicated to ,m4 ,liouville in the   
letter of date ,september #af1 #ahdf1   
referred to above1 but without proof1   
which ,thomson stated he had not time to
 write out owing to preparation for the 
commencement of his duties as ,professor
 of ,natural ,philosophy at ,glasgow on 
,november #a1 #ahdf4 ,it was not        
supplied until ,december #ahfh and      
,january #ahfi2 and in the meantime the 
problem had not been solved by any other
 mathematician4  ,as a starting point   
for this investigation the distribution 
on a thin plane circular disk of radius 
a is required4 ,this can be obtained by 
considering the disk as a limiting case 
of an oblate ellipsoid of revolution1   
charged to potential ,v1 say4 ,if ,fig4 
#aj represent the disk and ,p the point 
at which the density is sought1 so that 
,,cp  r1 and ,,ca  a1 the density is    

,v#b7;a  r74  ,the ratio q,v1 of charge 
to potential1 which is called the       
electrostatic capacity of the conductor1
 is thus #b;a1 that is a#a4ega4 ,it is1 
as ,thomson notes in his paper1 very    
remarkable that the ,hon4 ,henry        
,cavendish should have found long ago by
 experiment with the rudest apparatus   
the electrostatic capacity of a disk to 
be #aa4eg of that of a sphere of the    
same radius4  ,illustration3 ,,fig4 #aj4
  ,illustration3 ,,fig4 #aa4  ,now      
invert this disk distribution with any  
point ,q as centre of inversion1 and    
with radius of inversion a4 ,the        
geometrical inverse is a segment of a   
spherical surface which passes through  
,q4 ,the inverse distribution is the    
induced distribution on a conducting    
shell uninsulated and coincident with   
the segment1 and under the influence of 
a charge a,v situated at ,q 7,fig4 #aa74
 ,call this conducting shell the bowl4  
,if the surface-densities at            
corresponding points on the disk and on 

the inverse1 say points ,p and ,p'1 be s
 and s'1 then1 as on page #ea1 s'       
sa,,qp'4 ,if we put in the value of s   
given above1 that of s' can be put in a 
form given by ,thomson1 which it is     
important to remark is independent of   
the radius of the spherical surface4    
,this expression is applicable to the   
other side of the bowl1 inasmuch as the 
densities at near points on opposite    
sides of the plane disk are equal4  ,if 
v1 v' be the potentials at any point ,r 
of space1 due to the disk and to its    
image respectively1 v'  av,,qr4 ,if then
 ,r be coincident with a point ,p' on   
the spherical segment we have 7since    
then v  ,v7 ,v'  a,v,,qp'1 which is the 
potential due to the induced            
distribution caused by the charge a,v at
 ,q as already stated4  ,the fact that  
the value of s' does not involve the    
radius makes it possible to suppose the 
radius infinite1 in which case we have  
the solution for a circular disk        
uninsulated and under the influence of a

 charge of electricity at a point ,q in 
the same plane but outside the bounding 
circle4  ,now consider the two parts of 
the spherical surface1 the bowl ,b1 and 
the remainder ,s of the spherical       
surface4 ,q with the charge a,v may be  
regarded as situated on the latter part 
of the surface4 ,any other influencing  
charges situated on ,s will give        
distributions on the bowl to be found as
 described above1 and the resulting     
induced electrification can be found    
from these by summation4 ,if ,s be      
uniformly electrified to density s1 and 
held so electrified1 the inducing       
distribution will be one given by       
integration over the whole of ,s1 and   
the bowl ,b will be at zero potential   
under the influence of this             
electrification of ,s1 just as if ,b    
were replaced by a shell of metal       
connected to the earth by a long fine   
wire4 ,the densities are equal at       
infinitely near points on the two sides 
of ,b4  ,let the bowl be a thin metal   

shell connected with the earth by a long
 thin wire and be surrounded by a       
concentric and complete shell of        
diameter f greater than that of the     
spherical surface1 and let this shell be
 rigidly electrified with surface       
density s4 ,there will be no force      
within this shell due to its own        
electrification1 and hence it will      
produce no change of the distribution in
 the interior4 ,but the potential within
 will be #b;fs1 for the charge is fs1   
and the capacity of the shell is f4 ,the
 potential of the bowl will now be zero1
 and its electrification will just      
neutralise the potential -#b;fs1 that   
is1 will be exactly the free            
electrification required to produce     
potential #b;fs4  ,to find this         
electrification let the value of f be   
only infinitesimally greater than the   
diameter of the spherical surface of    
which ,b is a part2 then the bowl is    
under the influence 7#a7 of a uniform   
electrification of density s infinitely 

close to its outer surface1 and 7#b7 of 
a uniform electrification of the same   
density1 which may be regarded as upon  
the surface which has been called ,s    
above4 ,it is obvious that by 7#a7 a    
density s is produced on the outer      
surface of the bowl1 and no other       
effect2 by 7#b7 an equal density at     
infinitely near points on the opposite  
sides of the bowl is produced which we  
have seen how to calculate4 ,thus the   
distribution on the bowl freely         
electrified is completely determined and
 the density can easily be calculated4  
,the value will be found in ,thomson's  
paper4  ,interesting results are        
obtained by diminishing ,s more and more
 until the shell is a complete sphere   
with a circular hole in it4 ,tabulated  
results for different relative          
dimensions of ,s will be found in       
,thomson's paper1 ,reprint of ,papers1  
,articles ,v1 ,,xiv1 ,,xv4 ,also the    
reader will there find full particulars 
of the mathematical calculations        

indicated in this chapter1 and an       
extension of the method to the case of  
an influencing point not on the         
spherical surface of which the shell    
forms part4 ,further developments of the
 problem have been worked out by other  
writers1 and further information with   
references will be found in ,maxwell's  
,electricity and ,magnetism1 loc4 cit4  
,it is not quite clear whether ,thomson 
discovered geometrical inversion        
independently or not3 very likely he    
did4 ,his letter to ,liouville of date  
,october #h1 #ahde1 certainly reads as  
if he claimed the geometrical           
transformation as well as the           
application to electricity4 ,liouville1 
however1 in his ,note in which he dwells
 on the analytical theory of the        
transformation says1 ,la transformation 
dont il s'agit est bien connue1 du      
reste1 et des plus simples2 c'est celle 
que ,m4 ,thomson lui-mme a jadis employe
 sous le nom de principe des images4 ,in
 ,thomson and ,tail's ,natural          

,philosophy1  #eac1 the reference to the
 method is as follows3 ,irrespectively  
of the special electric application1 the
 method of images gives a remarkable    
kind of transformation which is often   
useful4 ,it suggests for mere geometry  
what has been called the transformation 
by reciprocal radius-rectors1 that is to
 say4444 ,then ,maxwell1 in his review  
of the ,reprint of ,papers 7,nature1    
vol4 vii71 after referring to the fact  
that the solution of the problem of the 
spherical bowl remained undemonstrated  
from #ahdf to #ahfi1 says that the      
geometrical idea of inversion had       
probably been discovered and            
rediscovered repeatedly1 but that in his
 opinion most of these discoveries were 
later than #ahde1 the date of ,thomson's
 first paper4#aj  ,a very general method
 of finding the potential at any point  
of a region of space enclosed by a given
 boundary was stated by ,green in his   
',essay' for the case in which the      
potential is known for every point of   

the boundary4 ,the success of the method
 depends on finding a certain function1 
now called ,green's function4 ,when this
 is known the potential at any point is 
at once obtained by an integration over 
the surface4 ,thomson's method of images
 amounts to finding for the case of a   
region bounded by one spherical surface 
or more the proper value of ,green's    
function4 ,green's method has been      
successfully employed in more           
complicated cases1 and is now a powerful
 method of attack for a large range of  
problems in other departments of        
physical mathematics4 ,thomson only     
obtained a copy of ,green's paper in    
,january #ahde1 and probably worked out 
his solutions quite independently of any
 ideas derived from ,green's general    
theory4      ,,chapter ,v  ,,the ,,chair
 ,,of ,,natural ,,philosophy ,,at       
,,glasgow4 ,,establishment ,,of ,,the   
,,first ,,physical ,,laboratory   ,the  
incumbent of the ,chair of ,natural     
,philosophy in the ,university of       

,glasgow1 ,professor ,meikleham1 had    
been in failing health for several      
years1 and from #ahdb to #ahde his      
duties had been discharged by another   
member of the ,thomson gens1 ,mr4 ,david
 ,thomson1 ,b4,a41 of ,trinity ,college1
 ,cambridge1 afterwards ,professor of   
,natural ,philosophy at ,aberdeen4 ,dr4 
,meikleham died in ,may #ahdf1 and the  
,faculty thereafter proceeded on the    
invitation of ,dr4 ,j4 ,p4 ,nichol1 the 
,professor of ,astronomy1 to consider   
whether in consequence of the great     
advances of physical science during the 
preceding quarter of a century it was   
not urgently necessary to remodel the   
arrangements for the teaching of natural
 philosophy in the ,university4 ,the    
advance of science had indeed been very 
great4 ,oersted and ,ampre1 ,henry and  
,faraday and ,regnault1 ,gauss and      
,weber1 had made discoveries and        
introduced quantitative ideas1 which had
 changed the whole aspect of            
experimental and mathematical physics4  

,the electrical discoveries of the time 
reacted on the other branches of natural
 philosophy1 and in no small degree on  
mathematics itself4 ,as a result the    
progress of that period has continued   
and has increased in rapidity1 until now
 the accumulated results1 for the most  
part already united in the grasp of     
rational theory1 have gone far beyond   
the power of any single man to follow1  
much less to master4  ,it is interesting
 to look into a course of lectures such 
as were usually delivered in the        
universities a hundred years ago by the 
,professor of ,natural ,philosophy4 ,we 
find a little discussion of mechanics1  
hydrostatics and pneumatics1 a little   
heat1 and a very little optics4         
,electricity and magnetism1 which in our
 day have a literature far exceeding    
that of the whole of physics only sixty 
years ago1 could hardly be said to      
exist4 ,the professor of the beginning  
of the nineteenth century1 when ,lord   
,kelvin's predecessor was appointed1    

apparently found himself quite free to  
devote a considerable part of each      
lecture to reflections on the beauties  
of nature1 and to rhetorical flights    
fitter for the pulpit than for the      
physics lecture-table4  ,in the         
intervening time the form and fashion of
 scientific lectures has entirely       
changed1 and the change is a testimony  
to the progress of science4 ,it is      
visible even in the design of the       
apparatus4 ,microscopes1 for example1   
have a perfection and a power undreamed 
of by our great-grandfathers1 and they  
are supported on stands which lack the  
ornamentation of that bygone time1 but  
possess stability and convenience4      
,everything and everybody--even the     
professor1 if that be possible--must be 
business-like2 and each moment of time  
must be utilised in experiments for     
demonstration1 not for applause1 and in 
brief and cogent statements of theory   
and fact4 ,to waste time in talk that is
 not to the point is criminal4 ,but     

withal there is need of grace of        
expression and vividness of description1
 of clearness of exposition1 of         
imagination1 even of poetical intuition3
 but the stern beauty of modern science 
is only disfigured by the old artificial
 adornments and irrelevancies4  ,this is
 the tone and temper of science at the  
present day3 the task is immense1 the   
time is short4 ,and sixty years since   
some tinge of the same cast of thought  
was visible in scientific workers and   
teachers4 ,the ,faculty agreed with ,dr4
 ,nichol that there was need to bring   
physical teaching and equipment into    
line with the state of science at the   
time2 but they wisely decided to do     
nothing until they had appointed a      
,professor of ,natural ,philosophy who  
would be able to advise them fully and  
in detail4 ,they determined1 however1 to
 make the appointment subject to such   
alterations in the arrangements of the  
department as they might afterwards find
 desirable4  ,on ,september #aa1 #ahdf1 

the ,faculty met1 and having considered 
the resolutions which had been proposed 
by ,dr4 ,nichol1 resolved to the effect 
that the appointment about to be made   
should not prejudice the right of the   
,faculty to originate or support1 during
 the incumbency of the new professor1   
such changes in the arrangements for    
conducting instruction in physical      
science as it might be expedient to     
adopt1 and that this resolution should  
be communicated to the candidate        
elected4 ,the minute then runs3 ,the    
,faculty having deliberated on the      
respective qualifications of the        
gentlemen who have announced themselves 
candidates for this chair1 and the vote 
having been taken1 it carried           
unanimously in favour of ,mr4 ,william  
,thomson1 ,b4,a41 ,fellow of ,st4       
,peter's ,college1 ,cambridge1 and      
formerly a student of this ,university1 
who is accordingly declared to be duly  
elected3 and ,mr4 ,thomson being within 
call appeared in ,faculty1 and the whole

 of this minute having been read to him 
he agreed to the resolution of ,faculty 
above recorded and accepted the office4 
,it was also resolved as follows3 ,the  
,faculty hereby prescribe ,mr4 ,thomson 
an essay on the subject1 ,de caloris    
distributione per terr corpus1 and      
resolve that his admission be on        
,tuesday the #acth ,october1 provided   
that he shall be found qualified by the 
,meeting and shall have taken the oath  
and made the subscriptions which are    
required by law4  ,at that time1 and    
down to within the last fifteen years1  
every professor1 before his induction to
 his chair1 had to submit a ,latin essay
 on some prescribed subject4 ,this was  
almost the last relic of the customs of 
the days when university lectures were  
delivered in ,latin1 a practice which   
appears to have been first broken       
through by ,adam ,smith when ,professor 
of ,moral ,philosophy4 ,whatever it may 
have been in the eighteenth century1 the
 ,latin essay at the end of the         

nineteenth was perhaps hardly an        
infallible criterion of the             
professor-elect's ,latinity1 and it was 
just as well to discard it4 ,but fifty  
years before1 and for long after1       
classical languages bulked largely in   
the curriculum of every student of the  
,scottish ,universities1 and it is      
undoubtedly the case that most of those 
who afterwards came to eminence in other
 departments of learning had in their   
time acquitted themselves well in the   
old ,litter ,humaniores4 ,this was true1
 as we have seen1 of ,thomson1 and it is
 unlikely that the form of his inaugural
 dissertation cost him much more effort 
than its matter4  ,illustration3        
,,professor ,,william ,,thomson1 #ahdf  
,the subject chosen had reference no    
doubt to the papers on the theory of    
heat which ,mr4 ,thomson had already    
published4 ,the thesis was presented to 
the ,faculty on the day appointed1 and  
approved1 and ,mr4 ,thomson having      
produced a certificate of his having    

taken the oaths to government1 and      
promised to subscribe the formula of the
 ,church of ,scotland as required by    
law1 on the first convenient            
opportunity1 the following oath was then
 administered to him1 which he took and 
subscribed3 ,ego1 ,gulielmus ,thomson1  
,b4,a41 physicus professor in hac       
,academia designatus1 promitto sancteque
 polliceor me in munere mihi demandato  
studiose fideliterque versaturum4       
,professor ,thomson was then solemnly   
admitted and received by all the        
,members present1 and took his seat as a
 ,member of ,faculty4  ,no translation  
of this essay was ever published1 but   
its substance was contained in various  
papers which appeared later4 ,the       
following reference to it is made in an 
introduction attached to ,article ,,xi  
of his ,mathematical and ,physical      
,papers 7vol4 i1 #ahhb74  ,an           
application to ,terrestrial             
,temperature1 of the principle set forth
 in the first part of this paper        

relating to the age of thermal          
distributions1 was made the subject of  
the author's ,inaugural ,dissertation on
 the occasion of his induction to the   
professorship of ,natural ,philosophy in
 the ,university of ,glasgow1 in        
,october #ahdf1 ',de ,motu ,caloris per 
,terr ,corpus'#aa3 which1 more fully    
developed afterwards1 gave a very       
decisive limitation to the possible age 
of the earth as a habitation for living 
creatures2 and proved the untenability  
of the enormous claims for ,,time which1
 uncurbed by physical science1          
geologists and biologists had begun to  
make and to regard as unchallengeable4  
,see ',secular ,cooling of the ,earth1  
,geological ,time1' and several other   
,articles below4 ,some statement of the 
argument for this limitation will be    
given later4 ,see ,chap4 ,,xiv4         
,thomson thus entered at the age of     
twenty-five on what was to be his life  
work as a teacher1 investigator1 and    
inventor4 ,for he continued in office   

fifty-three years1 so that the united   
tenures of his predecessor and himself  
amounted to only four years less than a 
century6 ,he took up his duties at the  
opening of the college session in       
,november1 and promptly called the      
attention of the ,faculty to the        
deficiencies of the equipment of        
apparatus1 which had been allowed to    
fall behind the times1 and required to  
have added to it many new instruments4  
,a committee was appointed to consider  
the question and report1 and as a result
 of the representations of this         
committee a sum of #ajj was placed at   
,professor ,thomson's disposal to supply
 his most pressing needs4 ,in the       
following years repeated applications   
for further grants were made and various
 sums were voted--not amounting to more 
than #ejj or #fjj in all--which were    
apparently regarded as 7and no doubt    
were1 considering the times and the     
funds at the disposal of the ,faculty7 a
 liberal provision for the teaching of  

physical science4 ,a minute of the      
,faculty1 of date ,nov4 #bf1 #ahdg1 is  
interesting4  ,after emphatically       
deprecating all idea that such large    
annual expenditure for any one          
department was to be regularly          
contemplated1 the committee refer in    
their report to the inadequate condition
 of the department in question1 and     
express their satisfaction with the     
reasonable manner in which the          
,professor of ,natural ,philosophy has  
on all occasions readily modified his   
demands in accordance with the          
economical suggestions of the committee4
 ,they conclude by saying that they view
 his ardour and anxiety in the          
prosecution of his profession with the  
greatest pleasure1 and heartily concur  
in those anticipations of his future    
celebrity which ,monsr4 ,serville1#ab   
the ,french mathematician1 has recently 
thought fit to publish to the scientific
 world4  ,again1 in ,april #aheb1 the   
,faculty agree to pay a sum of #acg #fs4

 #a;d4 as the price of purchases of     
philosophical apparatus already made1   
and approve of a suggestion of the      
committee that the expenditure on this  
behalf during the next year should not  
exceed #ej1 and they desire that the    
purchases shall be made so far as is    
possible with the previously obtained   
concurrence of the committee4 ,it is    
easy to imagine that the ardent young   
,professor of ,natural ,philosophy found
 the leisurely methods of his older     
colleagues much too slow1 and in his    
enthusiasm anticipated consent to his   
demands by ordering his new instruments 
without waiting for committees and      
meetings and reports4  ,in an address at
 the opening of the ,physical and       
,chemical ,laboratories of the          
,university ,college of ,north ,wales1  
on ,february #b1 #ahhe1 ,sir ,william   
,thomson 7as he was then7 referred to   
his early equipment and work as follows3
 ,when ,i entered upon the professorship
 of ,natural ,philosophy at ,glasgow1 ,i

 found apparatus of a very old-fashioned
 kind4 ,much of it was more than a      
hundred years old1 little of it less    
than fifty years old1 and most of it was
 worm-eaten4 ,still1 with such          
appliances1 year after year1 students of
 natural philosophy had been brought    
together and taught as well as possible4
 ,the principles of dynamics and        
electricity had been well illustrated   
and well taught1 as well taught as      
lectures and so imperfect apparatus--but
 apparatus merely of the                
lecture-illustration kind--could teach4 
,but there was absolutely no provision  
of any kind for experimental            
investigation1 still less idea1 even1   
for anything like students' practical   
work4 ,students' laboratories for       
physical science were not then thought  
of4#ac  ,it appears that the class of   
,natural ,philosophy 7there was then as 
a rule only one class in any subject1   
though supplementary work was done in   
various ways7 met for systematic        

lectures at #i a4m41 which is the hour  
still adhered to1 and for what was      
called ,experimental ,physics at #h     
p4m46  ,the ,university ,calendar for   
#ahfc-d states that the ,natural        
,philosophy ,class meets two hours      
daily1 #i a4m4 and #aa a4m4 ,the first  
hour is chiefly spent in statements of  
,principles1 description of ,results of 
,observation1 and ,experimental         
,illustrations4 ,the second hour is     
devoted to ,mathematical ,demonstrations
 and ,exercises1 and ,examinations on   
all parts of the ,course4  ,the ,text   
,books to be used are3 ',elements of    
,dynamics' 7first part now ready71      
,printed by ,george ,richardson1        
,university ,printer4 ',elements of     
,natural ,philosophy1' by ,professors   
,w4 ,thomson and ,p4 ,g4 ,tait 7,two    
,treatises to be published before       
,november4 ,macmillan4#ad7  ,the shorter
 of the last mentioned ,treatises will  
be used for the work required of all    
students of ,natural ,philosophy in the 

regular curriculum4 ,the whole or       
specified parts of the larger ,treatise 
will be prescribed in connection with   
voluntary examinations and exercises in 
the ,class1 and for candidates for the  
degree of ,m4,a4 with honours4 ,students
 who desire to undertake these higher   
parts of the business of the class1     
ought to be well prepared on all the    
subjects of the ,senior ,mathematical   
,class4  ,the ,laboratory in connection 
with the class is open daily from #i    
a4m4 to #d p4m4 for ,experimental       
,exercises and ,investigations1 under   
the direction of the ,professor and his 
official assistant4  ,in #ahdg the      
meetings for experimental physics were  
changed to #aa a4m4 ,the hour #i a4m4 is
 still 7#aijh7 retained for the regular 
meetings of the ordinary class1 and #aa 
a4m4 for meetings held twice a week for 
exercises and tutorial work1 attendance 
at which is optional4  ,a second        
graduating class has now been instituted
 and is very largely attended4 ,each    

student attends three lectures and      
spends four hours in the laboratory each
 week4 ,a higher class1 in two          
divisions1 is also held4  ,at an early  
date in his career as a professor       
,thomson called in the aid of his       
students for experimental research4 ,in 
many directions the properties of matter
 still lay unexplored1 and it was       
necessary to obtain exact data for the  
perfecting of the theories of           
elasticity1 electricity and heat1 which 
had been based on the researches of the 
first half of the nineteenth century4   
,to the authors of these                
theories--,gauss1 ,green1 ,cauchy and   
others--he was a fit successor4 ,not    
knowing all that had been done by these 
men of genius1 he reinvented1 as we have
 seen1 some of their great theorems1 and
 in somewhat later work1 notably in     
electricity and magnetism1 set the      
theories on a new basis cleared of all  
extraneous and unnecessary matter1 and  
reduced the hypotheses and assumptions  

to the smallest possible number1 stated 
with the most careful precautions       
against misunderstanding4 ,as this work 
was gradually accomplished the need for 
further experiment became more and more 
clearly apparent4 ,accordingly he       
established at the old ,college in the  
,high ,street1 what he has justly       
claimed was the first physical          
laboratory for students4#ae ,an old     
wine-cellar in the basement adjoining   
the ,natural ,philosophy ,class-room was
 first annexed1 and was the scene of    
early researches1 which were to lead to 
much of the best work of the present    
time4 ,to this was added a little later 
the ,blackstone ,examination-room1      
which1 disused and left unprotected1 was
 added to the wine-cellar1 and gave     
space for the increasing corps of       
enthusiastic workers who came under the 
influence of the new teacher1 and were  
eager to be associated with his work4 ,a
 good many of the researches which were 
carried out in this meagre accommodation

 in the old ,college will be mentioned  
in what follows4  ,illustration3 ,,inner
 ,,court ,,of ,,the ,,old ,,college     
,showing ,natural ,philosophy ,rooms    
,in the view of the inner court of the  
,old ,college given opposite1 the       
windows on the ground-floor to the right
 of the turret in front1 are those of   
the ,blackstone ,examination-room1 which
 formed a large part of the new         
,physical ,laboratory4 ,the windows     
above these1 on the second floor1 are   
those of the ,apparatus-room of the     
,natural ,philosophy ,department4       
,between the turret on the right of the 
picture and the angle of the court are  
the windows of the ,natural ,philosophy 
,class-room4 ,the attic above the       
,apparatus-room was at a later time     
occupied by the ,engineering            
,department1 under ,professor ,macquorn 
,rankine4  ,here again we may quote from
 the ,bangor address3  ,soon after ,i   
entered my present chair in the         
,university of ,glasgow in #ahdf ,i had 

occasion to undertake some              
investigations of electrodynamic        
qualities of matter1 to answer questions
 suggested by the results of            
mathematical theory1 questions which    
could only be answered by direct        
experiment4 ,the labour of observing    
proved too heavy1 much of it could      
scarcely be carried on without two or   
more persons1 working together4 ,i      
therefore invited students to aid in the
 work4 ,they willingly accepted the     
invitation1 and lent me most cheerful   
and able help4 ,soon after1 other       
students1 hearing that their            
class-fellows had got experimental work 
to do1 came to me and volunteered to    
assist in the investigation4 ,i could   
not give them all work in the particular
 investigation with which ,i had        
commenced--'the electric convection of  
heat'--for want of means and time and   
possibilities of arrangement1 but ,i did
 all in my power to find work for them  
on allied subjects 7,electrodynamic     

,properties of ,metals1 ,moduluses of   
,elasticity of ,metals1 ,elastic        
,fatigue1 ,atmospheric ,electricity1    
etc474 ,i then had an ordinary class of 
a hundred students1 of whom some        
attended lectures in natural philosophy 
two hours a day1 and had nothing more to
 do from morning till night4 ,these were
 the balmy days of natural philosophy in
 the ,university of ,glasgow--the       
pre-,commissional days4 ,but the        
majority of the class really had very   
hard work1 and many of them worked after
 class-hours for self-support4 ,some    
were engaged in teaching1 some were     
city-missionaries1 intending to go into 
the ,established ,church of ,scotland or
 some other religious denomination of   
,scotland1 or some of the denominations 
of ,wales1 for ,i always had many ,welsh
 students4 ,in those days1 as now1 in   
the ,scottish ,universities all         
intending theological students took a   
'philosophical curriculum'--'zuerst     
collegium logicum1' then moral          

philosophy1 and 7generally last7 natural
 philosophy4 ,three-fourths of my       
volunteer experimentalists used to be   
students who entered the theological    
classes immediately after the completion
 of the philosophical curriculum4 ,i    
well remember the surprise of a great   
,german professor when he heard of this 
rule and usage3 ',what6 do the          
theologians learn physics8' ,i said1    
',yes1 they all do2 and many of them    
have made capital experiments4 ,i       
believe they do not find that their     
theology suffers at all from 7their7    
having learned something of mathematics 
and dynamics and experimental physics   
before they enter upon it4'  ,this      
statement1 besides throwing an          
interesting light on the conditions of  
university work sixty years ago1 gives  
an illustration of the wide             
interpretation in ,scotland of the term 
,arts4 ,here it has meant1 since the    
,chair of ,natural ,philosophy was      
founded in #aegg1 and held by one of the

 ,regents of the ,university1 ,artes    
,liberales in the widest sense1 that is1
 the study of ,litter ,humaniores       
7including mental and moral philosophy7 
and physical and mathematical science4  
,these were all deemed necessary for a  
liberal education at that time3 in the  
scientific age in which we live it is   
more imperative than ever that neither  
should be excluded from the ,arts       
curriculum of our ,universities4 ,the   
common distinction between ,arts and    
,science is a false one1 and the product
 of a narrow idea which is alien to the 
traditions of our northern              
,universities4  ,it is to be noted1     
however1 that the laboratory thus       
founded was essentially a research      
laboratory2 it was not designed for the 
systematic instruction of students in   
methods of experimenting4 ,laboratories 
for this purpose came later1 and as a   
natural consequence4 ,but for the best  
students1 ill prepared as1 no doubt1    
some of them were for the work of       

research1 the experience gained in such 
a laboratory was very valuable4 ,they   
learned--and1 indeed1 had to learn--in  
an incidental manner how to determine   
physical constants1 such as specific    
gravities1 thermal capacities1 electric 
resistances1 and so forth4 ,for1 apart  
from the ,relations des ,expriences of  
,regnault1 and the magnetic and electric
 work of ,gauss and ,weber1 there was no
 systematised body of information       
available for the guidance of students4 
,good students could branch out from the
 main line of inquiry1 so as to acquire 
skill in subsidiary determinations of   
this kind2 to the more easily daunted   
student such difficulties proved        
formidable1 and often absolutely        
deterrent4  ,it is not easy for a       
physicist of the present day to realise 
the state of knowledge of the time1 and 
so he often fails to recognise the full 
importance of ,thomson's work4 ,the want
 of precise knowledge of physical       
constants was to a considerable extent a

 consequence of the want of exact       
definitions of quantities to be         
determined1 and in a much greater degree
 of the lack of any system of units of  
measurement4 ,the study of phenomena was
 in the main merely qualitative2 where  
an attempt had been made to obtain      
quantitative determinations1 the units  
employed were arbitrary and dependent on
 apparatus in the possession of the     
experimenter1 and therefore unavailable 
to others4 ,in the department of heat1  
as has been said1 a great beginning had 
been made by ,regnault1 in whose hands  
the exact determination of physical     
constants had become a fine art4  ,in   
electricity and magnetism there were    
already the rudiments of quantitative   
measurement4 ,but it was only long      
after1 when the actions of magnets and  
of electric currents had been much      
further studied1 that the ,british      
,association entered on its great work  
of setting up a system of absolute units
 for the measurement of such actions4   

,up till then the resistance1 for       
example1 of a piece of wire1 to the     
passage of an electric current along it1
 was expressed by some such             
specification as that it was equal to   
the resistance of a certain piece of    
copper wire in the experimenter's       
possession4 ,it was therefore           
practically impossible for experimenters
 elsewhere to profit by the information4
 ,and so in other cases4 ,an example    
from ,thomson's papers on the ,dynamical
 ,theory of ,heat may be cited here1    
though it refers to a time 7#ahea7 when 
some progress towards obtaining a system
 of absolute units had been made4 ,in   
#aah 7,art4 ,,xlviii7 he states that the
 electromotive force of a thermoelectric
 couple of copper and bismuth1 at       
temperatures #j ,c4 and #ajj ,c4 of its 
functions1 might be estimated from a    
comparison made by ,pouillet of the     
strength of the current sent by this    
electromotive force through a copper    
wire #bj metres long and #a millimetre  

in diameter1 with the strength of a     
current decomposing water at a certain  
rate1 were it not that the specific     
resistances of different specimens of   
copper are found to differ considerably 
from one another4 ,hence1 though an     
estimate is made1 it is stated that1    
without experiments on the actual wire  
used by ,pouillet1 it was impossible to 
arrive at an accurate result4 ,now if it
 had been in ,pouillet's power to       
determine accurately the resistance of  
his circuit in absolute units1 there    
would have been no difficulty in the    
matter1 and his result would have been  
immediately available for the estimate  
required4  ,when submarine cables came  
to be manufactured and laid all this had
 to be changed4 ,for they were          
expensive2 an ,atlantic cable1 for      
example1 cost half a million sterling4  
,the state of the cable had to be       
ascertained at short intervals during   
manufacture2 a similar watch had to be  
kept upon it during the process of      

laying1 and afterwards during its life  
of telegraphic use4 ,the observations   
made by one observer had therefore to be
 made available to all1 so that1 with   
other instruments and at another place1 
equivalent observations could be made   
and their results quantitatively        
compared with those of the former4 ,to  
set up a system of measurement for such 
purposes as these involved much         
theoretical discussion and an enormous  
amount of experimental investigation4   
,this was undertaken by a special       
committee of the ,association1 and a    
principal part in furnishing discussions
 of theory and in devising experimental 
methods was taken by ,thomson4 ,the     
committee's investigations took place at
 a date somewhat later in ,thomson's    
career than that with which we are here 
dealing1 and some account of them will  
be given in a later chapter2 but much   
work1 preparatory for and leading up to 
the determination of electrical         
standards1 was done by the volunteer    

laboratory corps in the transformed     
wine-cellar of the old ,college4  ,the  
selection and realisation of electrical 
standards was a work of extraordinary   
importance to the world from every point
 of view--political1 commercial1 and    
social4 ,it not only rendered           
applications of electricity possible in 
the arts and industries1 but by         
relieving experimental results from the 
vagueness of the specifications formerly
 in use1 made the further progress of   
pure electrical science a matter in     
which every step forward1 taken by an   
individual worker1 facilitated the      
advance of all4 ,but like other toilsome
 services1 the nature of which is not   
clear to the general public1 it has     
never received proper acknowledgment    
from those who have profited by it4 ,if 
,thomson had done nothing more than the 
work he did in this connection1 first   
with his students and later with the    
,british ,association ,committee1 he    
would have deserved well of his         

fellow-countrymen4  ,when ,professor    
,thomson was entering on the duties of  
his chair1 and calling his students to  
his aid1 the discoveries of ,faraday on 
the induction of currents by the motion 
of magnets in the neighbourhood of      
closed circuits of wire1 or1 what comes 
to the same thing1 the motion of such   
circuits in the fields of magnets1 had  
not been long given to the world1 and   
were being pondered deeply by natural   
philosophers4 ,the time was ripe for a  
quantitative investigation of current   
induction1 like that furnished by the   
genius of ,ampre after the discovery by 
,oersted of the deflection of a magnet  
by an electric current4 ,such an        
investigation was immensely facilitated 
by ,faraday's conception of lines of    
magnetic force1 the cutting of which by 
the wire of the circuit gave rise to the
 induced current4 ,indeed1 the          
mathematical ideas involved were        
indicated1 and not obscurely1 by        
,faraday himself4 ,but to render the    

mathematical theory explicit1 and to    
investigate and test its consequences1  
required the highest genius4 ,this work 
was accomplished in great measure by    
,thomson1 whose presentation of         
electrodynamic theory helped ,maxwell to
 the view that light was an affair of   
the propagation of electric and magnetic
 vibrations in an insulating medium1 the
 light-carrying ether4  ,another        
investigation on which he had already   
entered in #ahdg was of great           
importance1 not only for pure science   
but for the development and proper      
economy of all industrial operations4   
,the foundations on which a dynamical   
theory of heat was to be raised had been
 partly laid by ,carnot and were being  
completed on the experimental side by   
,james ,prescott ,joule1 whom ,thomson  
met in #ahdg at the meeting of the      
,british ,association at ,oxford4 ,the  
meeting at ,oxford in #ahfj is memorable
 to the public at large1 mainly on      
account of the discussion which took    

place on the ,darwinian theory1 and the 
famous dialectic encounter between      
,bishop ,wilberforce and ,professor     
,huxley2 the ,oxford meeting of #ahid   
will always be associated with the      
announcement of the discovery of argon  
by ,lord ,rayleigh and ,sir ,william    
,ramsay3 the meeting of #ahdg might     
quite as worthily be remembered as that 
at which ,joule laid down1 with         
numerical exactitude1 the first law of  
thermodynamics4 ,joule brought his      
experimental results before the         
,mathematical and ,physical ,section at 
that meeting2 and it appears probable   
that they would have received scant     
attention had not their importance been 
forcibly pointed out by ,thomson4       
,communications thereafter passed       
frequently between the two young        
physicists1 and there soon began a      
collaboration of great value to science1
 and a friendship which lasted till the 
death of ,joule in #ahhd4 ,see p4 #hh   
below4  ,we shall devote the next few   

chapters to an account1 as free from    
technicalities as possible1 of these    
great divisions of ,thomson's earlier   
original work as professor at ,glasgow4 
,,chapter ,,vi  ,,friendship ,,with     
,,stokes ,,and ,,joule4 ,,early ,,work  
,,at ,,glasgow   ,during his residence  
at ,cambridge ,thomson gained the       
friendship of ,george ,gabriel ,stokes1 
who had graduated as ,senior ,wrangler  
and ,first ,smith's ,prizeman in #ahda4 
,they discussed mathematical questions  
together and contributed articles on    
various topics to the ,cambridge        
,mathematical ,journal4 ,in #ahdf       
,cambridge and ,dublin was substituted  
for ,cambridge in the title of the      
,journal1 and a new series was begun    
under the editorship of ,thomson4 ,a    
feature of the earlier volumes of the   
new issue was a series of ,notes on     
,hydrodynamics written by agreement     
between ,thomson and ,stokes1 and       
printed in vols4 ii1 iii1 and v4 ,the   
first1 second1 and fifth of the series  

were written by ,thomson1 the others by 
,stokes4 ,the matter of these ,notes was
 not altogether novel2 but many points  
were put in a new and more truly        
physical light1 and the series was no   
doubt of much service to students1 for  
whose use the articles were intended4   
,some account of these ,notes will be   
given in a later chapter on ,thomson's  
hydrodynamical papers4  ,for the        
mathematical power and sure physical    
instinct of ,stokes ,thomson had always 
the greatest admiration4 ,when asked on 
one occasion who was the most           
outstanding worker in physical science  
on the continent1 he replied1 ,i do not 
know1 but whoever he is1 ,i am certain  
that ,stokes is a match for him4 ,in a  
report of an address which he delivered 
in ,june #ahig1 at the celebration of   
the ,jubilee of ,sir ,george ,stokes as 
,lucasian ,professor of ,mathematics1   
,lord ,kelvin referred to their early   
intercourse at ,cambridge in terms which
 were reported as follows3 ,when he     

reflected on his own early progress1 he 
was led to recall the great kindness    
shown to himself1 and the great value   
which his intercourse with ,sir ,george 
,stokes had been to him through life4   
,whenever a mathematical difficulty     
occurred he used to say to himself1     
',ask ,stokes what he thinks of it4' ,he
 got an answer if answer was possible2  
he was told1 at all events1 if it was   
unanswerable4 ,he felt that in his      
undergraduate days1 and he felt it more 
now4  ,after the death of ,stokes in    
,february #aijb1 ,lord ,kelvin again    
referred1 in an enthusiastic tribute in 
,nature for ,february #ab1 to these     
early discussions4 ,stokes's scientific 
work and scientific thought is but      
partially represented by his published  
writings4 ,he gave generously and freely
 of his treasures to all who were       
fortunate enough to have an opportunity 
of receiving from him4 ,his teaching me 
the principles of solar and stellar     
chemistry when we were walking about    

among the colleges sometime prior to    
#aheb 7when ,i vacated my ,peterhouse   
,fellowship to be no more in ,cambridge 
for many years7 is but one example4     
,the interchange of ideas between       
,stokes and ,thomson which began in     
those early days went on constantly and 
seems to have been stimulating to both4 
,the two men were in a sense            
complementary in nature and temperament4
 ,both had great power and great        
insight1 but while ,stokes was uniformly
 calm1 reflective1 and judicial1        
,thomson's enthusiasm was more          
outspokenly fervid1 and he was apt to be
 at times vehement and impetuous in his 
eagerness to push on an investigation2  
and though1 as became his nationality1  
he was cautious in committing himself to
 conclusions1 he exercised perhaps less 
reserve in placing his results before   
the public of science4  ,a              
characteristic instance of ,thomson's   
vehement pursuit of experimental results
 may be given here1 although the        

incidents occurred at a much later date 
in his career than that with which we   
are at present concerned4 ,in #ahhj the 
invention of the ,faure ,secondary      
,battery attracted his attention4 ,m4   
,faure brought from ,paris some cells   
made up and ready charged1 and showed in
 the ,physical ,laboratory at ,glasgow  
the very powerful currents which1 in    
consequence of their very low internal  
resistance1 they were capable of        
producing in a thick piece of copper    
wire4 ,the cells were of the original   
form1 constructed by coating strips of  
sheet lead on both sides with a paste of
 minium moistened with dilute sulphuric 
acid1 swathing them in woollen cloth    
sewed round them1 and then rolling two  
together to form the pair of plates for 
one cell4  ,a supply of sheet lead1     
minium1 and woollen cloth was at once   
obtained1 and the whole laboratory corps
 of students and staff was set to work  
to manufacture secondary batteries4 ,a  
small ,siemens-,halske dynamo was       

telegraphed for to charge the cells1 and
 the ventilating steam-engine of the    
,university was requisitioned to drive  
the dynamo during the night4 ,thus the  
,university stokers and engineer were   
put on double shifts2 the cells were    
charged during the night and the        
charging current and battery-potential  
measured at intervals4  ,then the cells 
were run down during the day1 and their 
output measured in the same way4 ,just  
as this began1 ,thomson was laid up with
 an ailment which confined him to bed   
for a couple of weeks or so2 but this   
led to no cessation of the laboratory   
activity4 ,on the contrary1 the         
laboratory corps was divided into two   
squads1 one for the night1 the other for
 the day1 and the work of charging and  
discharging1 and of measurement of      
expenditure and return of energy went on
 without intermission4 ,the results     
obtained during the day were taken to   
,thomson's bedside in the evening1 and  
early in the morning he was ready to    

review those which had been obtained    
during the night1 and to suggest further
 questions to be answered without delay4
 ,this mode of working could not go on  
indefinitely1 but it continued until his
 assistants 7some of whom had to take   
both shifts671 to say nothing of the    
stokers and students1 were fairly well  
exhausted4  ,on other occasions1 when he
 was from home1 he found the post too   
slow to convey his directions to his    
laboratory workers1 and telegraphed from
 day to day questions and instructions  
regarding the work on hand4 ,thus one   
important result 7anticipated1 however1 
by ,villari7 of the series of researches
 on the effects of stress on            
magnetisation which forms ,part ,,vii of
 his ,electrodynamic ,qualities of      
,metals--the fact that up to a certain  
magnetising force the effect of pull1   
applied to a wire of soft iron1 is to   
increase the magnetisation produced1 and
 for higher magnetising forces to       
diminish it--was telegraphed to him on  

the night on which the paper was read to
 the ,royal ,society4  ,it will thus be 
seen that ,thomson1 whether confined to 
his room or on holiday1 kept his mind   
fixed upon his scientific or practical  
work1 and was almost impatient for its  
progress4 ,stokes worked mainly by      
himself2 but even if he had had a corps 
of workers and assistants1 it is        
improbable that such disturbances of    
hours of attendance and laboratory and  
workshop routine would have occurred1 as
 were not infrequent at ,glasgow when   
,thomson's work was1 in the 'sixties and
 'seventies1 at its intensest4  ,stokes 
and ,thomson were in succession         
presidents of the ,royal ,society1      
,stokes from #ahhe to #ahij1 and        
,thomson 7from #ahib as ,lord ,kelvin7  
from #ahij to #ahie4 ,this is the       
highest distinction which any scientific
 man in this country can achieve1 and it
 is very remarkable that there should   
have been in recent times two presidents
 in succession whose modes of thought   

and mathematical power are so directly  
comparable with those of the great      
founder of modern natural philosophy4   
,stokes had the additional distinction  
of being the lineal successor of ,newton
 as ,lucasian ,professor of ,mathematics
 at ,cambridge4 ,but it was reserved for
 ,thomson to do much by the publication 
of ,thomson and ,tait's ,natural        
,philosophy to bring back the current of
 teaching and thought in dynamical      
science to the ideas of the ,principia1 
and to show how completely the          
fundamental laws1 as laid down in that  
great classic1 avail for the inclusion  
of the modern theory of energy1 in all  
its transformations1 within the category
 of dynamical action between material   
systems4  ,an exceedingly eminent       
politician1 now deceased1 said some     
years ago that the present age was      
singularly deficient in minds of the    
first quality4 ,so far as scientific    
genius is concerned1 the dictum was     
singularly false3 we have here a        

striking proof of the contrary4 ,but    
then few politicians know anything of   
science2 indeed some of those who guide1
 or aspire to guide1 the destinies of   
the most scientific and industrial      
empire the world has ever seen are      
almost boastful of their ignorance4     
,there are1 of course1 honourable       
exceptions4  ,it is convenient to refer 
here to the share which ,stokes and     
,thomson took in the physical           
explanation of the dark lines of the    
solar spectrum1 and to their prediction 
of the possibility of determining the   
constitution of the stars and of        
terrestrial substances by what is now   
known as spectrum analysis4 ,thomson    
used to give the physical theory of     
these lines in his lectures1 and say    
that he obtained the idea from ,stokes  
in a conversation which they had in the 
garden of ,pembroke at ,cambridge1 some 
time prior to #aheb 7see the quotation  
from his ,nature article quoted above1  
p4 #hj1 and the ,baltimore ,lectures1 p4

 #aja74 ,this is confirmed by a         
student's note-book1 of date #ahed1     
which is now in the ,natural ,philosophy
 ,department4 ,the statements therein   
recorded are perfectly definite and     
clear1 and show that at that early date 
the whole affair of spectrum analysis   
was in his hands1 and only required     
confirmation by experiments on the      
reversal of the lines of terrestrial    
substances by an atmosphere of the      
substance which produced the lines1 and 
a comparison of the positions of the    
bright lines of terrestrial substances  
with those of the dark lines of the     
solar spectrum4 ,why ,thomson did not   
carry out all these experiments it would
 be difficult to say4 ,some of them he  
did make1 for ,professor ,john          
,ferguson1 who was a student of ,natural
 ,philosophy in #ahei-fj1 has recently  
told how he witnessed ,thomson make the 
experiment of reversing the lines of    
sodium by passing the light from the    
salted flame of a spirit lamp through   

vapour of sodium produced by heating the
 metal in an iron spoon4 ,a few days    
later1 says ,professor ,ferguson1       
,thomson read a letter to his class     
announcing ,bunsen and ,kirchhoff's     
discovery4  ,a letter of ,stokes to ,sir
 ,john ,lubbock1 printed in the         
,scientific ,correspondence of ,sir     
,george ,gabriel ,stokes1 states his    
recollection of the matter1 and gives   
,thomson the credit of having inferred  
the method of spectrum analysis1 a      
method to which ,stokes himself makes no
 claim4 ,he says1 ,i know1 ,i think1    
what ,sir ,william ,thomson was alluding
 to4 ,i knew well1 what was generally   
known1 and is mentioned by ,herschel in 
his treatise on ,light1 that the bright 
,d seen in flames is specially produced 
when a salt of soda is introduced4 ,i   
connected it in my own mind with the    
presence of sodium1 and ,i suppose      
others did so too4 ,the coincidence in  
position of the bright and dark ,d is   
too striking to allow us to regard it as

 fortuitous4 ,in conversation with      
,thomson ,i explained the connection of 
the dark and bright line by the analogy 
of a set of piano strings tuned to the  
same note1 which1 if struck1 would give 
out that note1 and also would be ready  
to sound it1 to take it up1 in fact1 if 
it were sounded in air4 ,this would     
imply absorption of the arial           
vibrations1 as otherwise there would be 
a creation of energy4 ,accordingly ,i   
accounted for the presence of the dark  
,d in the solar spectrum by supposing   
that there was sodium in the atmosphere1
 capable of absorbing light of that     
particular refrangibility4 ,he asked me 
if there were any other instances of    
such coincidences of bright and dark    
lines1 and ,i said ,i thought there was 
one mentioned by ,brewster4 ,he was much
 struck with this1 and jumped to the    
conclusion that to find out what        
substances were in the stars we must    
compare the positions of the dark lines 
seen in their spectra with the spectra  

of metals1 etc4444  ,i should have said 
that ,i thought ,thomson was going too  
fast ahead1 for my notion at the time   
was that1 though a few of the dark lines
 might be traced to elementary          
substances1 sodium for one1 probably    
potassium for another1 yet the great    
bulk of them were probably due to       
compound vapours1 which1 like peroxide  
of nitrogen and some other known        
compound gases1 have the character of   
selective absorption4  ,it will be      
remembered that the experimental        
establishment of the method of spectrum 
analysis was published towards the end  
of #ahei by ,bunsen and ,kirchhoff1 to  
whom1 therefore1 the full credit of     
discoverers must be given4  ,lord       
,kelvin in the later years of his life  
used to tell the story of his first     
meeting with ,joule at ,oxford1 and of  
their second meeting a fortnight later  
in ,switzerland4 ,he did so also in his 
address delivered on the occasion of the
 unveiling of a statue of ,joule1 in    

,manchester ,town ,hall1 on ,december   
#g1 #ahic1 and we quote the narrative on
 account of its scientific and personal 
interest4 ,i can never forget the       
,british ,association at ,oxford in     
#ahdg1 when in one of the sections ,i   
heard a paper read by a very unassuming 
young man1 who betrayed no consciousness
 in his manner that he had a great idea 
to unfold4 ,i was tremendously struck   
with the paper4 ,i at first thought it  
could not be true1 because it was       
different from ,carnot's theory1 and    
immediately after the reading of the    
paper ,i had a few words with the       
author1 ,james ,joule1 which was the    
beginning of our forty years'           
acquaintance and friendship4 ,on the    
evening of the same day1 that very      
valuable institution of the ,british    
,association1 its conversazione1 gave us
 opportunity for a good hour's talk and 
discussion over all that either of us   
knew of thermodynamics4 ,i gained ideas 
which had never entered my mind before1 

and ,i thought ,i1 too1 suggested       
something worthy of ,joule's            
consideration when ,i told him of       
,carnot's theory4 ,then and there in the
 ,radcliffe ,library1 ,oxford1 we       
parted1 both of us1 ,i am sure1 feeling 
that we had much more to say to one     
another and much matter for reflection  
in what we had talked over that evening4
 ,but 444 a fortnight later1 when       
walking down the valley of ,chamounix1  
,i saw in the distance a young man      
walking up the road towards me1 and     
carrying in his hand something which    
looked like a stick1 but which he was   
using neither as an alpenstock nor as a 
walking-stick4 ,it was ,joule with a    
long thermometer in his hand1 which he  
would not trust by itself in the        
char--banc1 coming slowly up the hill   
behind him1 lest it should get broken4  
,but there1 comfortably and safely      
seated in the char--banc1 was his       
bride--the sympathetic companion and    
sharer in his work of after years4 ,he  

had not told me in ,section ,a1 or in   
the ,radcliffe ,library1 that he was    
going to be married in three days1 but  
now in the valley of ,chamounix he      
introduced me to his young wife4 ,we    
appointed to meet again a fortnight     
later at ,martigny to make experiments  
on the heat of a waterfall 7,sallanches7
 with that thermometer3 and afterwards  
we met again and again1 and from that   
time1 indeed1 remained close friends    
till the end of ,joule's life4 ,i had   
the great pleasure and satisfaction for 
many years1 beginning just forty years  
ago1 of making experiments along with   
,joule which led to some important      
results in respect to the theory of     
thermodynamics4 ,this is one of the most
 valuable recollections of my life1 and 
is indeed as valuable a recollection as 
,i can conceive in the possession of any
 man interested in science4  ,at the    
beginning of his course of lectures each
 session1 ,professor ,thomson read1 or  
rather attempted to read1 an            

introductory address on the scope and   
methods of physical science1 which he   
had prepared for his first session in   
#ahdf4 ,it set forth the fact that in   
science there were two stages of        
progress--a natural history stage and a 
natural philosophy stage4 ,in the first 
the discoverer or teacher is occupied   
with the collection of facts1 and their 
arrangement in classes according to     
their nature2 in the second he is       
concerned with the relations of facts   
already discovered and classified1 and  
endeavours to bring them within the     
scope of general principles or causes4  
,once the philosophical stage is        
reached1 its methods and results are    
connected and enlarged by continued     
research after facts1 controlled and    
directed by the conclusions of general  
theory4 ,thus the method is at first    
purely inductive1 but becomes in the    
second stage both inductive and         
deductive2 the general theory predicts  
by its deductions1 and the verification 

of these by experiment and observation  
give a validity to the theory which no  
mere induction could afford4 ,these     
stages of scientific investigation are  
well illustrated by the laws of ,kepler 
arrived at by mere comparison of the    
motions of the planets1 and the         
deduction of these laws1 with the       
remarkable correction of the third law1 
given by the theory of universal        
gravitation4 ,the prediction of the     
existence and place of the planet       
,neptune from the perturbations of      
,uranus is an excellent example of the  
predictive quality of a true            
philosophical theory4  ,the lecture then
 proceeded to state the province of     
dynamics1 to define its different parts1
 and to insist on the importance of     
kinematics1 which was described as a    
purely geometrical subject1 the geometry
 of motion1 considerations from which   
entered into every dynamical problem4   
,this distinction between dynamical and 
kinematical considerations--between     

those in which force is concerned and   
those into which enter only the idea of 
displacement in space and in time--is   
emphasised in ,thomson and ,tait's      
,natural ,philosophy1 which commences   
with a long chapter devoted entirely to 
kinematics4  ,whether ,professor        
,thomson read the whole of the          
,introductory ,lecture on the first     
occasion is uncertain--,clerk ,maxwell  
is said to have asserted that it was    
closely adhered to1 for that one time   
only1 and finished in much less than the
 hour allotted to it4 ,in later years he
 had never read more than a couple of   
pages when some new illustration1 or new
 fact of science1 which bore on his     
subject1 led him to digress from the    
manuscript1 which was hardly ever       
returned to1 and after a few minutes was
 mechanically laid aside and forgotten4 
,once on beginning the session he       
humorously informed the assembled class 
that he did not think he had ever       
succeeded in reading the lecture through

 before1 and added that he had          
determined that they should hear the    
whole of it6 ,but again occurred the    
inevitable digression1 in the           
professor's absorption in the new topic 
the promise was forgotten1 and the      
written lecture fared as before6 ,these 
digressions were exceedingly interesting
 to the best students3 whether they     
compensated for the want of a carefully 
prepared presentation of the elements of
 the subject1 suited to the wants of the
 mass of the members of the class1 is a 
matter which need not here be discussed4
 ,all through his elementary            
lectures--introductory or not--new ideas
 and new problems continually presented 
themselves4 ,an eminent physicist once  
remarked that ,thomson was perhaps the  
only living man who made discoveries    
while lecturing4 ,that was hardly true2 
in the glow of action and stress of     
expression the mind of every intense    
thinker often sees new relations1 and   
finds new points of view1 which amount  

to discoveries4 ,but fecundity of mind  
has1 of course1 its disadvantages3 the  
unexpected cannot happen without causing
 distractions to all concerned4 ,a mind 
which can see a theory of the physical  
universe in a smoke-ring is likely1     
unless kept under extraordinary and     
hampering restraint1 to be tempted to   
digress from what is strictly the       
subject in hand1 to the world of matters
 which that subject suggests4 ,professor
 ,thomson was1 it must be admitted1 too 
discursive for the ordinary student1 and
 perhaps did not study the art of       
boiling down physical theories to the   
form most easily digestible4 ,his       
eagerness of mind and width of mental   
outlook gave his lectures a special     
value to the advanced student1 so that  
there was a compensating advantage4     
,the teacher of natural philosophy is   
really placed in a position of          
extraordinary difficulty4 ,the fabric of
 nature is woven without seam1 and to   
take it to pieces is in a manner to     

destroy it4 ,it must1 after examination 
in detail1 be reconstructed and         
considered as a whole1 or its meaning   
escapes us4 ,and here lies the          
difficulty3 every bit of matter stands  
in relation to everything else1 and both
 sides of every relation must be        
considered4 ,in other words1 in the     
explanation of any one phenomenon the   
explanation of all others is more or    
less involved4 ,this does not mean that 
investigation or exposition is          
impossible1 or that we cannot proceed   
step by step2 but it shows the          
foolishness of that criticism of science
 and scientific method which asks for   
complete or ultimate knowledge1 and of  
the popular demand for a simple form of 
words to express what is in reality     
infinitely complex4  ,in the earlier    
years of his professorship ,professor   
,thomson taught his class entirely      
himself1 and gathered round him1 as he  
has told us in the ,bangor address1 an  
enthusiastic band of workers who aided  

him in the researches which he began on 
the electrodynamic qualities of metals1 
the elastic properties of substances1   
the thermal and electrical              
conductivities of metals1 and at a later
 date in the electric and magnetic work 
which he undertook as a member of the   
,british ,association ,committee on     
,electrical ,standards4 ,the class met1 
as has been stated1 twice a day1 first  
for lectures1 then for exercises and    
oral examination4 ,the changes which    
took place later in the curriculum1 and 
especially the introduction of honours  
classes in the different subjects1      
rendered it difficult1 if not           
impossible1 for two hours' attendance to
 be given daily on all subjects1 and    
students were at first excused          
attendance at the second hour1 and      
finally such attendance became          
practically optional4 ,but so long as   
the old traditional curriculum in       
,arts--of ,humanity1 ,greek1 ,logic1    
,mathematics1 ,moral ,philosophy and    

,natural ,philosophy--endured1 a large  
number of students found it profitable  
to attend at both hours1 and it was     
possible to give a large amount of      
excellent tutorial instruction by the   
working of examples and oral            
examination4  ,thomson always held that 
his commission included the subject of  
physical astronomy1 and though his      
lectures on that subject were1 as a     
rule1 confined to a statement of        
,kepler's laws and ,newton's deductions 
from them1 he took care that the written
 and oral examinations included         
astronomical questions1 for which the   
students were enjoined to prepare by    
reading ,herschel's ,outlines1 or some  
similar text-book4 ,this injunction not 
infrequently was disregarded1 and       
discomfiture of the student followed as 
a matter of course1 if he was called on 
to answer4 ,nor were the questions      
always easy to prepare for by reading4  
,a man might have a fair knowledge of   
elementary astronomy1 and be unable to  

answer offhand such a question as1 ,why 
is the ecliptic called the ecliptic8 or 
to say1 when the lectures on ,kepler had
 been omitted1 short and tersely just   
what was ,newton's deduction from the   
third law of the planetary motions4     
,home exercises were not prescribed as  
part of the regular work except from    
time to time in the ,higher             
,mathematical ,class which for thirty   
years or more of ,thomson's tenure of   
office was held in the department4 ,but 
the whole ordinary class met every      
,monday morning and spent the usual     
lecture hour in answering a paper of    
dynamical and physical questions4 ,as   
many as ten1 and sometimes eleven1      
questions were set in these papers1 some
 of them fairly difficult and involving 
novel ideas1 and by this weekly paper of
 problems the best students1 a dozen or 
more perhaps1 were helped to acquire a  
faculty of prompt and brief expression4 
,it was not uncommon for a good man to  
score #hj or #ij or even #ajj per cent4 

in the paper1 no small feat to          
accomplish in a single hour4 ,but to a  
considerable majority of the class1 it  
is doubtful whether the weekly          
examination was of much advantage3 they 
attempted one or two of the more        
descriptive questions perhaps1 but a    
good many did next to nothing4 ,the     
examinations came every week1 and so the
 preparation for one after another was  
neglected1 and as much procrastination  
of work ensued as there would have been 
if only four or five papers a session   
had been prescribed4 ,then the work of  
looking over so many papers was a heavy 
task to the professor's assistant1 a    
task which became impossible when1 for a
 few years in the early 'eighties1 the  
students in the ordinary class numbered 
about #bej4  ,the subject of natural    
philosophy had become so extensive in   
#ahdf that ,professor ,j4 ,p4 ,nichol   
called attention to the necessity for   
special arrangements for its adequate   
teaching4 ,what would he say if he could

 survey its dimensions at the present   
time6 ,to give even a brief outline of  
the principal topics in dynamics1 heat1 
acoustics1 light1 magnetism1 and        
electricity is more than can be         
accomplished in any course of university
 lectures2 and the only way to teach    
well and economically the large numbers 
of students#af who now throng the       
physics classes is to give each week1   
say1 three lectures as well considered  
and arranged as possible1 without any   
interruption from oral examination1 and 
assemble the students in smaller classes
 two or three times a week for exercises
 and oral examination4  ,thomson stated 
his views as to examinations and        
lectures in the ,bangor address4 ,the   
object of a university is teaching1 not 
testing1 444 in respect to the teaching 
of a university the object of           
examination is to promote the teaching4 
,the examination should be1 in the first
 place1 daily4 ,no professor should meet
 his class without talking to them4 ,he 

should talk to them and they to him4    
,the ,french call a lecture a confrence1
 and ,i admire that idea4 ,every lecture
 should be a conference of teachers and 
students4 ,it is the true ideal of a    
professorial lecture4 ,i have found that
 many students are afflicted when they  
come up to college with the disease     
called 'aphasia4' ,they will not answer 
when questioned1 even when the very     
words of the answer are put in their    
mouths1 or when the answer is simply    
'yes' or 'no4' ,that disease wears off  
in a few weeks1 but the great cure for  
it is in repeated and careful and very  
free interchange of question and answer 
between teacher and student4444 ,written
 examinations are very important1 as    
training the student to express with    
clearness and accuracy the knowledge he 
has gained1 but they should be once a   
week to be beneficial4  ,the great      
difficulty now1 when both classes and   
subject have grown enormously1 is to    
have free conversation between professor

 and student1 and yet give an adequate  
account of the subject4 ,to examine     
orally in a thorough way two students in
 each class-hour is about as much as can
 be done if there is to be any          
systematic exposition by lecture at all2
 and thus the conference between teacher
 and individual student can occur only  
twice a year at most4 ,nevertheless     
,lord ,kelvin was undoubtedly right3    
oral examination and the training of    
individual students in the art of clear 
and ready expression are very desirable4
 ,the real difficulties of the subject  
are those which occur to the best       
students1 and a discussion of them in   
the presence of others is good for all4 
,this is difficult nowadays1 for large  
classes cannot afford to wait while two 
or three backward students grope after  
answers to questions--which in many     
cases must be on points which are       
sufficiently plain to the majority--to  
say nothing of the temptation to        
disorder which the display of personal  

peculiarities or oddities of expression 
generally affords to an assembly of     
students4 ,but time will be economised  
and many advantages added1 if large     
classes are split up into sections for  
tutorial work1 to supplement the careful
 presentation of the subject made in the
 systematic lectures delivered to the   
whole class in each case4 ,the          
introduction of a tutorial system will1 
however1 do far more harm than good1    
unless the method of instruction is such
 as to foster the self-reliance of the  
student1 who must not be1 so to speak1  
spoon-fed3 such a method1 and the       
advantages of the weekly examination on 
paper may be secured1 by setting the    
tutorial class to work out on the spot  
exercises prescribed by the lecturer4   
,but the danger1 which is a very real   
one1 can only be fully avoided by the   
precautions of a skilful teacher1 who in
 those small classes will draw out and  
direct the ideas of his students1 rather
 than impart knowledge directly4  ,after

 a few years ,thomson found it necessary
 to appoint an assistant1 and ,mr4      
,donald ,mc,farlane1 who had            
distinguished himself in the            
,mathematics and ,natural ,philosophy   
classes1 was chosen4 ,mr4 ,mc,farlane   
was originally a block-printer1 and     
seems to have been an apprentice at     
,alexandria in the ,vale of ,leven1 at  
the time of the passing of the first    
,reform ,bill4 ,after some time spent in
 the cotton industry of the district1 he
 became a teacher in a village school in
 the ,vale of ,leven1 and afterwards    
entered the ,university as a student4   
,he discharged his duties in the most   
faithful and self-abnegating manner     
until his retirement in #ahhj1 when he  
had become advanced in years4 ,he had   
charge of the instruments of the        
department1 got ready the lecture       
illustrations and attended during       
lecture to assist in the experiments and
 supply numerical data when required1   
prepared the weekly class examination   

paper and read the answers handed in1   
and assisted in the original            
investigations which the professor was  
always enthusiastically pursuing4 ,a    
kind of universal physical genius was   
,mc,farlane2 an expert calculator and an
 exact and careful experimentalist4     
,many a long and involved arithmetical  
research he carried out1 much apparatus 
he made in a homely way1 and much he    
repaired and adjusted4 ,then1 always    
when the professor was out of the way   
and calm had descended on the           
apparatus-room1 if not on the           
laboratory1 ,mc,farlane sat down to     
reduce his pile of examination papers1  
lest ,monday should arrive with a new   
deluge of crude answers and queer       
mistakes1 ere the former had            
disappeared4 ,on ,friday afternoons at  
#c o'clock he gave solutions of the     
previous ,monday's questions to any     
members of the class who cared to       
attend2 and his clear and deliberate    
explanations were much appreciated4 ,an 

unfailing tribute was rendered to him   
every year by the students1 and often   
took the form of a valuable gift for    
which one and all had subscribed4 ,a    
recluse he was in his way1 hardly       
anybody knew where he lived--the        
professor certainly did not--and a man  
of the highest ability and of the most  
absolute unselfishness4 ,an hour in the 
evening with one or two special friends1
 and the study of ,german1 were the only
 recreations of ,mc,farlane's solitary  
life4 ,he was full of humour1 and told  
with keen enjoyment stories of the      
,university worthies of a bygone age4   
,for thirty years he worked on for a    
meagre salary1 for during the earlier   
part of that time no provision for      
assistants was made in the ,government  
grant to the ,scottish ,universities4   
,by an ordinance issued in #ahfa by the 
,university ,commissioners1 appointed   
under the ,act of #aheh1 a grant of #ajj
 a year was made from the ,consolidated 
,fund for an assistant in each of the   

departments of ,humanity1 ,greek1       
,mathematics1 and ,natural ,philosophy1 
and for two in the department of        
,chemistry2 and ,mc,farlane's position  
was somewhat improved4 ,his veneration  
for ,thomson was such as few students or
 assistants have had for a master3 his  
devotion resembled that of the old      
famulus rather than the much more       
measured respect paid by modern         
assistants to their chiefs4  ,after his 
retirement ,mc,farlane lived on in      
,glasgow1 and amused himself reading    
out-of-the-way ,latin literature and    
with the calculation of eclipses6 ,he   
finally returned to ,alexandria1 where  
he died in ,february #ahig4 ,old        
,mc,farlane will be held in affectionate
 remembrance so long as students of the 
,natural ,philosophy ,class in the      
'fifties and 'sixties and 'seventies1   
now1 alas6 a fast vanishing band1       
survive4  ,soon after taking his degree 
of ,b4,a4 at ,cambridge in #ahde1       
,thomson had been elected a ,fellow of  

,st4 ,peter's ,college4 ,in #aheb he    
vacated his ,fellowship on his marriage 
to ,miss ,margaret ,crum1 daughter of   
,mr4 ,walter ,crum of ,thornliebank1    
near ,glasgow1 but was re-elected in    
#ahga1 and remained thereafter a ,fellow
 of ,peterhouse throughout his life4    
,,chapter ,,vii  ,,the ,,account ,,of   
,,carnot',s ,,theory ,,of ,,the ,,motive
 ,,power ,,of ,,heat--,,transition ,,to 
,,the ,,dynamical ,,theory ,,of ,,heat  
,the meeting of ,thomson and ,joule at  
,oxford in #ahdg was fraught with       
important results to the theory of heat4
 ,thomson had previously become         
acquainted with ,carnot's essay1 most   
probably through ,clapeyron's account of
 it in the ,journal de l'cole           
,polytechnique1 #ahcd1 and had adopted  
,carnot's view that when work was done  
by a heat engine heat was merely let    
down from a body at one temperature to a
 body at a lower temperature4 ,joule    
apparently knew nothing of ,carnot's    
theory1 and had therefore come to the   

consideration of the subject without any
 preconceived opinions4 ,he had thus    
been led to form a clear notion of heat 
as something which could be transformed 
into work1 and vice versa4 ,this was the
 root idea of his attempt to find the   
dynamical equivalent of heat4 ,it was   
obvious that a heat engine took heat    
from a source and gave heat to a        
refrigerator1 and ,joule naturally      
concluded that the appearance of the    
work done by the engine must be         
accompanied by the disappearance of a   
quantity of heat of which the work done 
was the equivalent4 ,he carried this    
idea consistently through all his work  
upon energy-changes1 not merely in heat 
engines but in what might be called     
electric engines4 ,for he pointed out   
that the heat produced in the circuit of
 a voltaic battery was the equivalent of
 the energy-changes within the battery1 
and that1 moreover1 when an             
electromagnetic engine was driven by the
 current1 or when electrochemical       

decomposition was effected in a         
voltameter in the circuit1 the heat     
evolved in the circuit for a given      
expenditure of the materials of the     
battery was less than it would otherwise
 have been1 by the equivalent of the    
work done by the engine1 or of the      
chemical changes effected in the        
voltameter4 ,thus ,joule was in         
possession at an earlier date than      
,thomson of the fundamental notion upon 
which the true dynamical theory of heat 
engines is founded4 ,thomson1 on the    
other hand1 as soon as he had received  
this idea1 was able to add to it the    
conception1 derived from ,carnot1 of a  
reversible engine as the engine of      
greatest efficiency1 and to deduce in a 
highly original manner all the          
consequences of these doctrines which go
 to make up the ordinary thermodynamics 
even of the present time4 ,though       
,clausius was the first1 as we shall    
see1 to deduce various important        
theorems1 yet ,thomson's discussion of  

the question had a quality peculiarly   
its own4 ,it was marked by that freedom 
from unstated assumptions1 from         
extraneous considerations1 from         
vagueness of statement and of thought1  
which characterises all his applications
 of mathematics to physics4 ,the        
physical ideas are always set forth     
clearly and in such a manner that their 
quantitative representation is          
immediate3 we shall have an example of  
this in the doctrine of absolute        
temperature4 ,in most of the            
thermodynamical discussions which take  
the great memoir of ,clausius as their  
starting point1 temperature is supposed 
to be given by a hypothetical something 
which is called a perfect gas1 and it is
 very difficult1 if not impossible1 to  
gather a precise notion of the          
properties of such a gas and of the     
temperature scale thereon founded4      
,thomson's scale enables a perfect gas  
to be defined1 and the deviations of the
 properties of ordinary gases from those

 of such a gas to be observed and       
measured4  ,the idea1 then1 which ,joule
 had communicated to ,section ,a1 when  
,thomson interposed to call attention to
 its importance1 was that work spent in 
overcoming friction had its equivalent  
in the heat produced1 that1 in fact1 the
 amount of heat generated in such a case
 was proportional to the work spent1    
quite irrespective of the materials used
 in the process1 provided no change of  
the internal energy of any of them took 
place so as to affect the resulting     
quantity of heat4 ,this forced upon     
physicists the view pointed to by the   
doctrine of the immateriality of heat1  
established by the experiments of       
,rumford and ,davy1 that heat itself was
 a form of energy2 and thus the         
principle of conservation of energy was 
freed from its one defect1 its apparent 
failure when work was done against      
friction4  ,rumford had noted the very  
great evolution of heat when gun-metal  
was rubbed by a blunt borer1 and had    

come to the reasonable conclusion that  
what was evolved in apparently unlimited
 quantity by the abrasion or cutting    
down of a negligible quantity of        
materials could not be a material       
substance4 ,he had also made a rough    
estimate of the relation between the    
work spent in driving the borer by      
horse-power and the heat generated4     
,joule's method of determining the      
work-equivalent of heat was a refinement
 of ,rumford's1 but differed in the     
all-important respect that accurate     
means were employed for measuring the   
expenditure of work and the gain of     
heat4 ,he stirred a liquid1 such as     
water or mercury1 in a kind of churn    
driven by a falling weight4 ,the range  
of descent of the weight enabled the    
work consumed to be exactly estimated1  
and a sensitive thermometer in the      
liquid measured the rise of temperature2
 thus the heat produced was accurately  
determined4 ,the rise of temperature was
 very slight1 and the change of state of

 the liquid1 and therefore any possible 
change in its internal energy1 was      
infinitesimal4 ,the experiments were    
carried out with great care1 and        
included very exact measurements of the 
various corrections--for example1 the   
amount of work spent at pulleys and     
pivots without affecting the liquid1 and
 the loss of heat by radiation4 ,the    
experiments proved that the work spent  
on the liquid and the heat produced were
 in direct proportion to one another4   
,he found1 finally1 in #ahej1 that #ggb 
foot-pounds of work at ,manchester      
generated one ,british thermal unit1    
that is1 as much heat as sufficed to    
raise a pound of water from #fj ,f4 to  
#fa ,f4 ,an approximation to this       
conclusion was contained in the paper   
which he communicated to the ,british   
,association at ,oxford in #ahdg4  ,the 
results of a later determination made   
with an improved apparatus1 and         
completed in #ahgh1 gave a very slightly
 higher result4 ,when corrected to the  

corresponding ,fahrenheit degree on the 
air thermometer it must be increased by 
somewhat less than one per cent4 ,the   
exact relation has been the subject     
during the last twenty years of much    
refined experimental work1 but without  
any serious alteration of the number    
indicated above4  ,it is probable that  
in consequence of the conference which  
he had with ,joule at ,oxford ,thomson  
had his thoughts turned for some time   
almost exclusively to the dynamical     
theory of heat engines4 ,he worked at   
the subject almost continuously for a   
long time1 sending paper after paper to 
the ,edinburgh ,royal ,society4 ,as we  
have seen1 he had given ,joule a        
description of ,carnot's essay on the   
,motive ,power of ,heat and the         
conclusions1 or some of them1 therein   
contained4 ,joule's result1 and the     
thermodynamic law which it established1 
gave the key to the correction of       
,carnot's theory necessary to bring it  
into line with a complete doctrine of   

energy1 which should take account of    
work done against frictional            
resistances4  ,mayer of ,heilbronn had  
endeavoured to determine the dynamical  
equivalent of heat in #ahdb1 by         
calculating from the knowledge available
 at the time of the two specific heats  
of air--the specific heat at constant   
pressure and the specific heat at       
constant volume--the heat value of the  
work spent in compressing air from a    
given volume to a smaller one4 ,the     
principle of this determination is      
easily understood1 but it involves an   
assumption that is not always clearly   
perceived4 ,let the air be imagined     
confined in a cylinder closed by a      
frictionless piston1 which is kept from 
moving out under the air pressure by    
force applied from without4 ,let heat be
 given to the air so as to raise its    
temperature1 while the piston moves out 
so as to keep the pressure constant4 ,if
 the pressure be p and the increase of  
volume be dv1 the work done against the 

external force is pdv4 ,let the rise of 
temperature be one degree of the        
,centigrade scale1 and the mass of air  
be one gramme1 the heat given to the gas
 is the specific heat ,cp of the gas at 
constant pressure1 for there is only    
slight variation of specific heat with  
temperature4 ,but if the piston had been
 fixed the heat required for the same   
rise of temperature would have been ,cv1
 the specific heat at constant volume4  
,now ,mayer assumed that the excess of  
the specific heat ,cp above ,cv was the 
thermal equivalent of the work pdv done 
in the former case4 ,thus he obtained   
the equation ,j7,cp  ,cv7  pdv1 where ,j
 denotes the dynamical equivalent of    
heat and ,cp1 ,cv are taken in thermal  
units4 ,but if a be the coefficient of  
expansion of the air under constant     
pressure 7that is #abgc71 and v be the  
volume of the air at #j ,c41 we have dv 
av1 so that ,j7,cp  ,cv7  apv4 ,now if p
 be one atmosphere1 say #a4jad  #ajf    
dynes per square centimetre1 and the    

temperature be the freezing point of    
water1 the volume of a gramme of air is 
#a4jjabic in cubic centimetres4 ,hence  
,j7,cp  ,cv7  7#a4jad  #ajf77bgc        
4#jjabic7  from which1 if ,cp  ,cv is   
known1 the value of ,j can be found4    
,in ,mayer's time the difference of the 
specific heats of air was imperfectly   
known1 and so ,j could not be found with
 anything like accuracy4 ,from          
,regnault's experiments on the specific 
heat at constant pressure1 and from the 
known ratio of the specific heats as    
deduced from the velocity of sound      
combined with ,regnault's result1 the   
value of ,cp  ,cv may be taken as       
4#jfhf4 ,thus ,j works out to #db4b     
#ajf1 in ergs per calorie1 which is not 
far from the true value4 ,mayer obtained
 a result equivalent to #cf4e  #ajf ergs
 per calorie4  ,the assumption on which 
this calculation is founded is that     
there is no alteration of the internal  
energy of the gas in consequence of     
expansion4 ,if the air when raised in   

temperature1 and at the same time       
increased in volume1 contained less     
internal energy than when simply heated 
without alteration of volume1 the energy
 evolved would be available to aid the  
performance of the work done against    
external forces1 and less heat would be 
required1 or1 in the contrary case1 more
 heat would be required1 than would be  
necessary if the internal energy        
remained unaltered4 ,thus putting d,w   
for pdv1 the work done1 e for the       
internal energy before expansion1 and   
d,h for the heat given to the gas1 we   
have obviously the equation    ,jd,h  de
  d,w  where de is the change of        
internal energy due to the alteration of
 volume1 together with the alteration of
 temperature4 ,if now the temperature be
 altered without expansion1 no external 
work is done and d,w for that case is   
zero4 ,let e and ,h be the energy change
 and the heat supplied1 then in this    
case    ,j,h  e  ,o  ,thus    ,j7d,h    
,h7  de  e  d,w  and the assumption is  

that de  e1 so that d,w  ,j7d,h  ,h72   
that is1 d,w  ,j7,cp  ,cv71 when the    
rise of temperature is #a ,c4 and the   
mass of air is one gramme4 ,this        
assumption requires justification1 and  
by an experiment of ,joule's1 which was 
repeated in a more sensitive form       
devised by ,thomson1 it was shown to be 
a very close approximation to the truth4
 ,joule's experiment is well known3 the 
explanation given above may serve to    
make clear the nature of the research   
undertaken later by ,thomson and ,joule 
conjointly4  ,the inverse process1 the  
conversion of heat into work1 required  
investigation1 and it is this that      
constitutes the science of              
thermodynamics4 ,it was the subject of  
the celebrated ,rflexions sur la        
,puissance ,motrice du ,feu1 et sur les 
,machines ,propres  ,dvelopper cette    
,puissance1 published in #ahbd by ,sadi 
,carnot1 an uncle of the late ,president
 of the ,french ,republic4 ,only a few  
copies of this essay were issued1 and   

its text was known to very few persons  
twenty-four years later1 when it was    
reprinted by the ,academy of ,sciences4 
,its methods and conclusions were set   
forth by ,thomson in #ahdi in a memoir  
which he entitled1 ,an ,account of      
,carnot's ,theory of the ,motive ,power 
of ,heat4 ,numerical results deduced    
from ,regnault's experiments on steam   
were included2 and the memoir as a whole
 led naturally in ,thomson's hands to a 
corrected theory of heat engines1 which 
he published in #aheb4 ,carnot's view of
 the working of a heat engine was       
founded on the analogy of the           
performance of work by a stream of water
 descending from a higher level to a    
lower4 ,the same quantity of water flows
 away in a given time from a water wheel
 in the tail-race as is received in that
 time by the wheel from the supply      
stream4 ,now a heat engine receives heat
 from a supplying body1 or source1 at   
one temperature and parts with heat to  
another body 7for example1 the condenser

 of a steam engine7 at a lower          
temperature4 ,this body is usually      
called the refrigerator4 ,according to  
,carnot these temperatures corresponded 
to the two levels in the case of the    
water wheel2 the heat was what flowed   
through the engine4 ,thus in his theory 
as much heat was given up by a heat     
engine to the body at the lower         
temperature as was received by it from  
the source4 ,the heat was simply        
transferred from the body at the higher 
temperature to the body at the lower2   
and this transference was supposed to be
 the source of the work4#ag  ,the first 
law of thermodynamics based on ,joule's 
proportionality of heat produced to work
 expended1 and the converse assumed and 
verified a posteriori1 showed that this 
view is erroneous1 and that the heat    
delivered to the refrigerator must be   
less in amount than that received from  
the source1 by exactly the amount which 
is converted into work1 together with   
the heat which1 in an imperfect engine1 

is lost by conduction1 etc41 from the   
cylinder or other working chamber4 ,this
 change was made by ,thomson in his     
second paper3 but he found the ideas of 
,carnot of direct and fruitful          
application in the new theory4 ,these   
were the cycle of operations and the    
ideal reversible engine4  ,in the       
,carnot cycle the working               
substance--which might be a gas or a    
vapour1 or a liquid1 or a vapour and its
 liquid in contact3 it did not matter   
what for the result--was supposed to be 
put through a succession of changes in  
which the final state coincided with the
 initial4 ,thus the substance having    
been brought back to the same physical  
condition as it had when the cycle      
began1 has the same internal energy as  
it had at the beginning1 and in the     
reckoning of the work done by or against
 external forces1 nothing requires to be
 set to the account of the working      
substance4 ,this is the first great     
advantage of the method of reasoning    

which ,carnot introduced4  ,the ideal   
engine was a very simple affair3 but the
 notion of reversibility is difficult to
 express in a form sufficiently definite
 and precise4 ,carnot does not attempt  
this2 he merely contents himself with   
describing certain cycles of operations 
which obviously can be carried through  
in the reverse order4 ,nor does ,thomson
 go further in his ,account of ,carnot's
 ,theory1 though he states the criterion
 of a perfect engine in the words1 ,a   
perfect thermodynamic engine is such    
that1 whatever amount of mechanical     
effect it can derive from a certain     
thermal agency1 if an equal amount be   
spent in working it backwards1 an equal 
reverse thermal effect will be produced4
 ,this proposition was proved by        
,carnot3 and the following formal       
statement in the essay is made3 ,la     
puissance motrice de la chaleur est     
independante des agents mis en uvre pour
 la raliser3 sa quantit est fixe        
uniquement par les temperatures des     

corps entre lesquels se fait1 en dernier
 rsultat1 le transport du calorique4    
,the result involved in each1 that the  
work done in a cycle by an ideal engine 
depends on the temperatures between     
which it works and not at all on the    
working substance1 is1 as we shall see1 
of the greatest importance4 ,the proof  
of the proposition1 by supposing a more 
efficient engine than the ideal one to  
exist1 and to be coupled with the       
latter1 so that the more efficient would
 perform the cycle forwards and the     
ideal engine the same cycle backwards1  
is well known4 ,in ,carnot's view the   
former would do more work by letting    
down a given quantity of heat from the  
higher to the lower temperature than was
 spent on the latter in transferring the
 same quantity of heat from the lower to
 the higher temperature1 so that no heat
 would be taken from or given to source 
or refrigerator1 while there would be a 
gain of work on the whole4 ,this would  
be equivalent to admitting that useful  

work could be continually performed     
without any resulting thermal or other  
change in the agents performing the     
work4 ,even at that time this could not 
be admitted as possible1 and hence the  
supposition that a more efficient engine
 than the reversible one could exist was
 untenable4  ,carnot showed that the    
work done by an ideal engine1 in        
transferring heat from one temperature  
to another1 was to be found by means of 
a certain function of the temperature1  
hence called ,carnot's function4 ,the   
corresponding function in the true      
dynamical theory is always called       
,carnot's4 ,a certain assignment of     
value to it gave1 as we shall see1      
,thomson's famous absolute thermodynamic
 scale of temperature4  ,in the light of
 the facts and theories which now exist1
 and are almost the commonplaces of     
physical text books1 it is very         
interesting to review the ideas and     
difficulties which occurred to the      
founders of the science of heat sixty   

years ago4 ,for example1 ,thomson asks1 
in his ,account of ,carnot's ,theory1   
what becomes of the mechanical effect   
which might be produced by heat which is
 transferred from one body to another by
 conduction4 ,the heat leaves one body  
and enters another and no mechanical    
effect results3 if it passed from one to
 the other through a heat engine1       
mechanical effect would be produced3    
what is produced in place of the        
mechanical effect which is lost8 ,this  
he calls a very perplexing question1 and
 hopes that it will1 before long1 be    
cleared up4 ,he states1 further1 that   
the difficulty would be entirely avoided
 by abandoning ,carnot's principle that 
mechanical effect is obtained by the    
transference of heat from one body to   
another at a lower temperate4 ,joule    
urges precisely this solution of the    
difficulty in his paper1 ,on the        
,changes of ,temperature produced by the
 ,rarefaction and ,condensation of ,air 
7,phil4 ,mag41 ,may #ahde74 ,thomson    

notes this1 but adds1 ,if we do so1     
however1 we meet with innumerable other 
difficulties--insuperable without       
further experimental investigation1 and 
an entire reconstruction of the theory  
of heat from its foundation4 ,it is in  
reality to experiment that we must look1
 either for a verification of ,carnot's 
axiom1 and an explanation of the        
difficulty we have been considering1 or 
for an entirely new basis of the ,theory
 of ,heat4  ,the experiments here asked 
for had already1 as was soon after      
perceived by ,thomson1 been made by     
,joule1 not merely in his determinations
 of the dynamical equivalent of heat1   
but in his exceedingly important        
investigation of the energy changes in  
the circuit of a voltaic cell1 or of a  
magneto-electric machine4 ,moreover1 the
 answer to this very perplexing question
 was afterwards to be given by ,thomson 
himself in his paper1 ,on a ,universal  
,tendency in ,nature to the ,dissipation
 of ,mechanical ,energy1 published in   

the ,edinburgh ,proceedings in #aheb4   
,again1 we find1 a page or two earlier  
in the ,account of ,carnot's ,theory1   
the question asked with respect to the  
heat evolved in the circuit of a        
magneto-electric machine1 ,is the heat  
which is evolved in one part of the     
closed conductor merely transferred from
 those parts which are subject to the   
inducing influence8 and the statement   
made that ,joule had examined this      
question1 and decided that it must be   
answered in the negative4 ,but ,thomson 
goes on to say1 ,before we can finally  
conclude that heat is absolutely        
generated in such operations1 it would  
be necessary to prove that the inducing 
magnet does not become lower in         
temperature and thus compensate for the 
heat evolved in the conductor4  ,here1  
apparently1 the idea of work done in    
moving the magnet1 or the conductor in  
the magnetic field1 is not present to   
,thomson's mind2 for if it had been1 the
 idea that the work thus spent might    

have its equivalent1 in part1 at least1 
in heat generated in the circuit1 would 
no doubt have occurred to him and been  
stated4 ,this idea had been used just a 
year before by ,helmholtz1 in his essay 
,die ,erhaltung der ,kraft1 to account  
for the heat produced in the circuit by 
the induced current1 that is1 to answer 
the first question put above in the     
sense in which ,joule answered it4 ,the 
subject1 however1 was fully worked out  
by ,thomson in a paper published in the 
,philosophical ,magazine for ,december  
#ahea1 to which we shall refer later4   
,tables of the work performed by various
 steam engines working between different
 stated temperatures were given at the  
close of the ,account1 and compared with
 the theoretical duty as calculated for 
,carnot's ideal perfect engine4 ,of     
course the theoretical duty was         
calculated from the temperatures of the 
boiler and condenser2 the much greater  
fall of temperature from the furnace to 
the boiler was neglected as inevitable1 

so that the loss involved in that fall  
is not taken account of4 ,carnot's      
theory gave for the theoretical duty of 
one heat unit 7equivalent to #acij      
foot-pounds of work7 #ddj foot-pounds   
for boiler at #adj ,c4 and condenser at 
#cj ,c42 and the best performance       
recorded was #bec foot-pounds1 giving a 
percentage of #eg4e per cent4 ,the worst
 was that of common engines consuming   
#ab lb4 of coal per horse-power per     
hour1 and gave #ch4a foot-pounds1 or a  
percentage of #h4f per cent4 ,these     
percentages become on the dynamical     
theory #fh and #aj4c1 since the true    
theoretical duty for the heat unit is   
only #cga foot-pounds4  ,it is worthy of
 notice that the indicator-diagram      
method of graphically representing the  
changes in a cycle of operations is     
adopted in ,thomson's ,account1 but does
 not occur in ,carnot's essay4 ,the     
cycles consist of two isothermal changes
 and two adiabatic changes2 that is1 two
 changes at the temperatures of the     

source and refrigerator respectively1   
and two changes--from the higher to the 
lower temperature1 and from the lower to
 the higher4 ,these changes are made    
subject to the condition in each case   
that the substance neither gains nor    
loses energy in the form of heat1 but is
 cooled in the one case by expansion and
 heated in the other by compression4    
,the indicator diagram was due not to   
,thomson but to ,clapeyron 7see p4 #ii  
above71 who used it to illustrate an    
account of ,carnot's theory4  ,there    
appeared in the issue of the ,edinburgh 
,philosophical ,transactions for        
,january #b1 #ahdi1 along with the      
,account of ,carnot's ,theory1 a paper  
by ,james ,thomson1 entitled1           
,theoretical ,considerations on the     
,effect of ,pressure in ,lowering the   
,freezing ,point of ,water4 ,the author 
predicted that1 unless the principle of 
conservation of energy was at fault1 the
 effect of increase of pressure on water
 in the act of freezing would be to     

lower the freezing point2 and he        
calculated from ,carnot's theory the    
amount of lowering which would be       
produced by a given increment of        
pressure4 ,the prediction thus made was 
tested by experiments carried out in the
 ,physical ,laboratory by ,thomson1 and 
the results obtained completely         
confirmed the conclusions arrived at by 
theory4 ,this prediction and its        
verification have been justly regarded  
as of great importance in the history of
 the dynamical theory of heat2 and they 
afford an excellent example of the      
predictive character of a true          
scientific theory4 ,the theory of the   
matter will be referred to in the next  
chapter4     ,,chapter ,,viii           
,,thermodynamics ,,and ,,absolute       
,,thermometry   ,the first statement of 
the true dynamical theory of heat1 based
 on the fundamental idea that the work  
done in a ,carnot cycle is to be        
accounted for by an excess of the heat  
received from the source over the heat  

delivered to the refrigerator1 was given
 by ,clausius in a paper which appeared 
in ,poggendorff's ,annalen in ,march and
 ,april #ahej1 and in the ,philosophical
 ,magazine for ,july #ahej1 under a     
title which is a ,german translation of 
that of ,carnot's essay4 ,in that paper 
the ,first ,law of ,thermodynamics is   
explicitly stated as follows3 ,in all   
cases in which work is produced by the  
agency of heat1 a quantity of heat      
proportional to the amount of work      
produced is expended1 and1 inversely1 by
 the expenditure of that amount of work 
exactly the same amount of heat is      
generated4 ,modern thermodynamics is    
based on this principle and on the      
so-called ,second ,law of               
,thermodynamics2 which is1 however1     
variously stated by different authors4  
,according to ,clausius1 who used in his
 paper an argument like that of ,carnot 
based on the transference of heat from  
the source to the refrigerator1 the     
foundation of the second law was the    

fact that heat tends to pass from hotter
 to colder bodies4 ,in #ahed 7,pogg4    
,ann41 ,dec4 #ahed7 he stated his       
fundamental principle explicitly in the 
form3 ,heat can never pass from a colder
 to a hotter body1 unless some other    
change1 connected therewith1 take place 
at the same time1 and gives in a note   
the shorter statement1 which he regards 
as equivalent3 ,heat cannot of itself   
pass from a colder to a hotter body4    
,we shall not here discuss the manner in
 which ,clausius applied this principle3
 but he arrived at and described in his 
paper many important results1 of which  
he must therefore be regarded as the    
primary discoverer4 ,his theory as      
originally set forth was lacking in     
clearness and simplicity1 and was much  
improved by additions made to it on its 
republication1 in #ahfd1 with other     
memoirs on the ,theory of ,heat4  ,in   
the ,transactions ,r4,s4,e41 for ,march 
#ahea1 ,thomson published his great     
paper1 ,on the ,dynamical ,theory of    

,heat4 ,the object of the paper was     
stated to be threefold3 7#a7 ,to show   
what modifications of ,carnot's         
conclusions are required1 when the      
dynamical theory is adopted3 7#b7 ,to   
indicate the significance in this theory
 of the numerical results deduced from  
,regnault's observations on steam3 7#c7 
,to point out certain remarkable        
relations connecting the physical       
properties of all substances established
 by reasoning analogous to that of      
,carnot1 but founded on the dynamical   
theory4  ,this paper1 though subsequent 
to that of ,clausius1 is very different 
in character4 ,many of the results are  
identical with those previously obtained
 by ,clausius1 but they are reached by a
 process which is preceded by a clear   
statement of fundamental principles4    
,these principles have since been the   
subject of discussion1 and are not free 
from difficulty even now2 but a great   
step in advance was made by their       
careful formulation in ,thomson's paper1

 as a preliminary to the erection of the
 theory and the deduction of its        
consequences4 ,two propositions are     
stated which may be taken as the ,first 
and ,second ,laws of ,thermodynamics4   
,one is equivalent to the ,first ,law as
 stated in p4 #aaf1 the other enunciates
 the principle of ,reversibility as a   
criterion of perfection of a heat       
engine4 ,we quote these propositions4   
,prop4 ,i 7,joule74--,when equal        
quantities of mechanical effect are     
produced by any means whatever from     
purely thermal sources1 or lost in      
purely thermal effects1 equal quantities
 of heat are put out of existence or are
 generated4  ,prop4 ,,ii 7,carnot and   
,clausius74--,if an engine be such that 
when worked backwards1 the physical and 
mechanical agencies in every part of its
 motions are all reversed1 it produces  
as much mechanical effect as can be     
produced by any thermodynamic engine1   
with the same temperatures of source and
 refrigerator1 from a given quantity of 

heat4  ,prop4 ,i was proved by assuming 
that heat is a form of energy and       
considering always the work effected by 
causing a working substance to pass     
through a closed cycle of changes1 so   
that there was no change of internal    
energy to be reckoned with4  ,prop4 ,,ii
 was proved by the following axiom3 ,it 
is impossible1 by means of inanimate    
material agency1 to derive mechanical   
effect from any portion of matter by    
cooling it below the temperature of the 
coldest of the surrounding objects4     
,this is rather a postulate than an     
axiom2 for it can hardly be contended   
that it commands assent as soon as it is
 stated1 even from a mind which is      
conversant with thermal phenomena4 ,it  
sets forth clearly1 however1 and with   
sufficient guardedness of statement1 a  
principle which1 when the process by    
which work is done is always a cyclical 
one1 is not found contradicted by       
experience1 and one1 moreover1 which can
 be at once explicitly applied to       

demonstrate that no engine can be more  
efficient than a reversible one1 and    
that therefore the efficiency of a      
reversible engine is independent of the 
nature of the working substance4  ,it   
has been suggested by ,clerk ,maxwell   
that this axiom is contradicted by the  
behaviour of a gas4 ,according to the   
kinetic theory of gases an elevation of 
temperature consists in an increase of  
the kinetic energy of the translatory   
motion of the gaseous particles2 and no 
doubt there actually is1 from time to   
time1 a passage of some more quickly    
moving particles from a portion of a gas
 in which the average kinetic energy is 
low1 to a region in which the average   
kinetic energy is high1 and thus a      
transference of heat from a region of   
low temperature to one of higher        
temperature4 ,maxwell imagined a space  
filled with gas to be divided into two  
compartments ,a and ,b by a partition in
 which were small massless trapdoors1 to
 open and shut which required no        

expenditure of energy4 ,at each of these
 doors was stationed a sorting demon1   
whose duty it was to allow every        
particle having a velocity greater than 
the average to pass through from ,a to  
,b1 and to stop all those of smaller    
velocity than the average4 ,similarly1  
the demons were to prevent all quickly  
moving particles from going across from 
,b to ,a1 and to pass all slowly moving 
particles4 ,in this way1 without the    
expenditure of work1 all the quickly    
moving particles could be assembled in  
one compartment1 and all the slowly     
moving particles in the other2 and thus 
a difference of temperatures between the
 two compartments could be brought      
about1 or a previously existing one     
increased by transference of heat from a
 colder to a hotter mass of gas4        
,contrary to a not uncommon belief1 this
 process does not invalidate ,thomson's 
axiom as he intended it to be           
understood4 ,for the gas referred to    
here is what he would have regarded as  

the working substance of the engine1 by 
the cycles of which all the mechanical  
effect was derived2 and it is not1 at   
the end of the process1 in the state as 
regards average kinetic energy of the   
particles in which it was at first4     
,that this was his answer to the implied
 criticism of his axiom contained in    
,maxwell's illustration1 those who have 
heard him refer to the matter in his    
lectures are well aware4 ,but of course 
it is to be understood that the         
substance returns to the same state only
 in a statistical sense4  ,thomson's    
demonstration that a reversible engine  
is the most efficient is well known1 and
 need not here be repeated in detail4   
,the reversible engine may be worked    
backwards1 and the working substance    
will take in heat where in the direct   
action it gave it out1 and vice versa3  
the substance will do work against      
external forces where in the direct     
action it had work done upon it1 and    
vice versa3 in short1 all the physical  

and mechanical changes will be of the   
same amount1 but merely reversed1 at    
every stage of the backward process4    
,thus if an engine ,a be more efficient 
than a reversible one ,b1 it will       
convert a larger percentage of an amount
 of heat ,h taken in at the source into 
work than would the reversible one      
working between the same temperatures4  
,thus if h be the heat given to the     
refrigerator by ,a1 and h' that given by
 ,b when both work directly and take in 
,h2 h must be less than h'4 ,then couple
 the engines together so that ,b works  
backwards while ,a works directly4 ,a   
will take in ,h and deliver h1 and do   
work equivalent to ,h  h4 ,b will take  
h' from the refrigerator and deliver ,h 
to the source1 and have work equivalent 
to ,h  h' spent upon it4 ,there will be 
no heat on the whole given to or taken  
from the source2 but heat h'  h will be 
taken from the refrigerator1 and work   
equivalent to this will be done4 ,thus  
by a cyclical process1 which leaves the 

working substance as it was1 work is    
done at the expense of heat taken from  
the refrigerator1 which ,thomson's      
postulate affirms to be impossible4     
,therefore the assumption that an engine
 more efficient than the reversible     
engine exists must be abandoned2 and we 
have the conclusion that all reversible 
engines are equally efficient4  ,thomson
 acknowledged in his paper the priority 
of ,clausius in his proof of this       
proposition1 but stated that this       
demonstration had occurred to him before
 he was aware that ,clausius had dealt  
with the matter4 ,he now cited1 as      
examples of the ,first ,law of          
,thermodynamics1 the results of ,joule's
 experiments regarding the heat produced
 in the circuits of magneto-electric    
machines1 and the fact that when an     
electric current produced by a thermal  
agency or by a battery drives a motor1  
the heat evolved in the circuit by the  
passage of the current is lessened by   
the equivalent of the work done on the  

motor4  ,illustration3 ,,fig4 #ab4  ,in 
the ,carnot cycle1 the first operation  
is an isothermal expansion 7,,ab in     
,fig4 #ab71 in which the substance      
increases in volume by dv1 and takes in 
from the source heat of amount ,mdv4    
,the second operation is an adiabatic   
expansion1 ,,bc1 in which the volume is 
further increased and the temperature   
sinks by dt to the temperature of the   
refrigerator4 ,the third operation is an
 isothermal compression1 ,,cd1 until the
 volume and pressure are such that an   
adiabatic compression ,,da will just    
bring the substance back to the original
 state4 ,if pt be the rate of increase  
of pressure with temperature when the   
volume is constant1 the step of pressure
 from one isothermal to the other is    
pt4dt2 and thus the area of the closed  
cycle in the diagram which measures the 
external work done in the succession of 
changes is pt4dtdv4 ,now1 by the second 
law1 the work done must be a certain    
fraction of the work-equivalent of the  

heat1 ,mdv1 taken in from the source4   
,this fraction is independent of the    
nature of the working substance1 but    
varies with the temperature1 and is     
therefore a function of the temperature4
 ,its ratio to the difference of        
temperature dt between source and       
refrigerator was called ,carnot's       
function1 and the determination of this 
function by experiment was at first     
perhaps the most important problem of   
thermodynamics4 ,denoting it by 1 we    
have the equation    pt  ,m             
444 7,a7  which may be taken as         
expressing in mathematical language the 
second law of thermodynamics4 ,m is here
 so chosen that ,mdv is the heat        
expressed in units of work1 so that     
does not involve ,joule's equivalent of 
heat4 ,this equation was given by       
,carnot3 it is here obtained by the     
dynamical theory which regards the work 
done as accounted for by disappearance1 
not transference merely1 of heat4  ,the 
work done in the cycle becomes now      

,mdtdv1 or if ,h denote ,mdv1 it is     
,hdt4 ,the fraction of the heat utilised
 is thus dt4 ,this is called the        
efficiency of the engine for the cycle4 
,from the first law ,thomson obtained   
another fundamental equation4 ,for every
 substance there is a relation          
connecting the pressure p 7or more      
generally the stress of some type71 the 
volume v 7or the configuration according
 to the specified stress71 and the      
temperature4 ,we may therefore take     
arbitrary changes of any two of these   
quantities3 the relation referred to    
will give the corresponding change of   
the third4 ,thomson chose v and t as the
 quantities to be varied1 and supposed  
them to sustain arbitrary small changes 
dv and dt in consequence of the passage 
of heat to the substance from without4  
,the amount of heat taken in is ,mdv    
,ndt1 where ,mdv and ,ndt are heats     
required for the changes taken          
separately4 ,but the substance expanding
 through dv does external work pdv4     

,thus the net amount of energy given to 
the substance from without is ,mdv  ,ndt
  pdv or 7,m  p7dv  ,ndt2 and if the    
substance is made to pass through a     
cycle of changes so that it returns to  
the physical state from which it        
started1 the whole energy received in   
the cycle must be zero4 ,from this it   
follows that the rate of variation of ,m
  p when the temperature but not the    
volume varies1 is equal to the rate of  
variation of ,n when the volume but not 
the temperature varies4 ,to see that    
this relation holds1 the reader         
unacquainted with the properties of     
perfect differentials may proceed thus4 
,let the substance be subjected to the  
infinitesimal closed cycle of changes   
defined by 7#a7 a variation consisting  
of the simultaneous changes dv1 dt of   
volume and temperature1 7#b7 a variation
 dv of volume only1 7#c7 a variation dt 
of temperature only4 ,m  p and ,n vary  
so as to have definite values for the   
beginning and end of each step1 and the 

proper mean values can be written down  
for each step at once1 and therefore the
 value of 7,m  p7dv  ,ndt obtained4     
,adding together these values for the   
three steps we get the integral for the 
cycle4 ,the condition that this should  
vanish is at once seen to be the        
relation stated above4  ,this result    
combined with the equation ,a derived   
from the second law1 gives an important 
expression for ,carnot's function4  ,we 
shall not pursue this discussion        
further3 so much is given to make clear 
how certain results as to the physical  
properties of substances were obtained1 
and to explain ,thomson's scale of      
absolute thermodynamic temperature1     
which is by far the most important      
discovery within the range of           
theoretical thermodynamics4  ,there are 
several scales of temperature3 in point 
of fact the scale of a mercury-in-glass 
thermometer is defined by the process of
 graduation1 and therefore there are as 
many such scales as there are           

thermometers1 since no two specimens of 
glass expand in precisely the same way4 
,equal differences of temperature do not
 correspond to equal increments of      
volume of the mercury3 for the glass    
envelope expands also and in its own    
way4 ,on the scale of a constant        
pressure gas thermometer changes of     
temperature are measured by variations  
of volume of the gas1 while the pressure
 is maintained constant2 on a constant  
volume gas thermometer changes of       
temperature are measured by alterations 
of pressure while the volume of the gas 
is kept constant4 ,each scale has its   
own independent definition1 thus if the 
pressure of the gas be kept constant1   
and the volume at temperature #j ,c4 be 
v and that at any other temperature be v
 we define the numerical value t1 this  
latter temperature1 by the equation v   
v7#a  ,et71 where ,e is #aajj of the    
increase of volume sustained by the gas 
in being raised from #j ,c4 to #ajj ,c4 
,these are the temperatures of reference

 on an ordinary centigrade thermometer1 
that is1 the temperature of melting ice 
and of saturated steam under standard   
atmospheric pressure1 respectively4     
,thus t has the value 7vv  #a7;,e1 and  
is the temperature 7on the constant     
pressure scale of the gas thermometer7  
corresponding to the volume v4 ,equal   
differences of temperature are such as  
correspond to equal increments of the   
volume at #j ,c4  ,similarly1 on the    
constant volume scale we obtain a       
definition of temperature from the      
pressure p1 by the equation t  7pp      
#a7;,e'1 where p is the pressure at #j  
,c41 and ,e' is #aajj of the change of  
pressure produced by raising the        
temperature from #j ,c4 to #ajj ,c4     
,for air ,e is approximately #abgc1 and 
thus t  #bgc7v  v7v4 ,if we take the    
case of v  #j1 we get t  #bgc4 ,now1    
although this temperature may be        
inaccessible1 we may take it as zero1   
and the temperature denoted by t is1    
when reckoned from this zero1 #bgc  t4  

,this zero is called the absolute zero  
on the constant pressure air            
thermometer4 ,the value of ,e' is very  
nearly the same as that of ,e2 and we   
get in a similar manner an absolute zero
 for the constant volume scale4 ,if the 
gas obeyed ,boyle's law exactly at all  
temperatures1 ,e would not differ from  
,e'4  ,it was suggested to ,thomson by  
,joule1 in a letter dated ,december #i1 
#ahdh1 that the value of  might be given
 by the equation   ,,je7#a  ,et74 ,here 
we take heat in dynamical units1 and    
therefore the factor ,j is not required4
 ,with these units ,joule's suggestion  
is that   ,e7#a  ,et71 or with ,e  #abgc
   #a7bgc  t71 that is1   #a,t where ,t 
is the temperature reckoned in          
centigrade degrees from the absolute    
zero of the constant pressure air       
thermometer4  ,the possibility of       
adopting this value of  was shown by    
,thomson to depend on whether or not the
 heat absorbed by a given mass of gas in
 expanding without alteration of        

temperature is the equivalent of the    
work done by the expanding gas against  
external pressure4 ,the heat ,h absorbed
 by the air in expanding from volume ,v 
to another volume ,v' at constant       
temperature is the integral of ,mdv     
taken from the former volume to the     
latter4 ,but by the value of ,m given on
 p4 #aba1 if ,w be the integral of pdv1 
that is the work done by the air in the 
expansion1 ,wt  ,h4 ,the equation       
fulfilled by the gas at constant        
pressure 7the defining equation for t71 
v  v7#a  ,et71 gives for the integral of
 pdv1 that is ,w1 the equation ,w  pv7#a
  ,et7log7,v',v71 so that ,wt  ,,ew7#a  
,et74 ,thus ,h  ,,ew7#a  ,et74  ,hence  
it follows that if   ,e7#a  ,et71 the   
value of ,h will be simply ,w4 ,thus    
,joule's suggested value of  is only    
admissible if the work done by the gas  
in expanding from a given volume to any 
other is the equivalent of the heat     
absorbed2 or1 which is the same thing1  
if the external work done in compressing

 the gas from one volume to another is  
the equivalent of the heat developed4   
,this result naturally suggests the     
formation of a new scale of thermometry 
by the adoption of the defining relation
 ,t  #a1 where ,t denotes temperature4  
,a scale of temperature thus defined is 
proposed in the paper by ,joule and     
,thomson1 ,on the ,thermal ,effects of  
,fluids in ,motion1 ,part ,,ii1 which   
was published in the ,philosophical     
,transactions for ,june #ahed1 and is   
what is now universally known as        
,thomson's scale of absolute            
thermodynamic temperature4 ,it can1 of  
course1 be made to give #ajj as the     
numerical value of the temperature      
difference between #j ,c4 and #ajj ,c4  
by properly fixing the unit of ,t4 ,this
 scale was the natural successor1 in the
 dynamical theory1 of one which ,thomson
 had suggested in #ahdh1 and which was  
founded1 according to ,carnot's idea1 on
 the condition that a unit of heat      
should do the same amount of work in    

descending through each degree4 ,this1  
as he pointed out1 might justly be      
called an absolute scale1 since it would
 be independent of the physical         
properties of any substance4 ,in the    
same sense the scale defined by ,t  #a  
is truly an absolute scale4  ,the new   
scale gives a simple expression for the 
efficiency of a perfect engine working  
between two physically given            
temperatures1 and assigns the numerical 
values of these temperatures2 for the   
heat ,h taken in from the source in the 
isothermal expansion which forms the    
first operation of the cycle 7p4 #abj7  
is ,mdv1 and1 as we have seen1 the work 
done in the cycle is pt4dtdv1 or ,hdt4  
,if we adopt the expression #a,t for 1  
we may put d,t for dt2 and we obtain for
 the work done the expression ,hd,t,t4  
,the work done is thus the fraction     
d,t,t of the heat taken in1 and this is 
what is properly called the efficiency  
of the engine for the cycle4  ,if we    
suppose the difference of temperatures  

between source and refrigerator to be   
finite1 ,t  ,t'1 say1 then since ,t is  
the temperature of the source1 we have  
for the efficiency 7,t  ,t'7,t4 ,if the 
heat taken in be ,h1 the heat rejected  
is ,,ht',t1 so that the heat received by
 the engine is to the heat rejected by  
it in the ratio of ,t' to ,t4 ,thus1 as 
was done by ,thomson1 we may define the 
temperatures of the source and          
refrigerator as proportional to the heat
 taken in from the source and the heat  
rejected to the refrigerator by a       
perfect engine1 working between those   
temperatures4 ,the scale may be made to 
have #ajj degrees between the           
temperature of melting ice and the      
boiling point1 as already explained4 ,we
 shall return to the comparison of this 
scale with that of the air thermometer4 
,at present we consider some of the     
thermodynamic relations of the          
properties of bodies arrived at by      
,thomson4  ,first we take the working   
substance of the engine as consisting of

 matter in two states or phases2 for    
example1 ice and water1 or water and    
saturated steam4 ,let us apply equation 
7,a7 to this case4 ,if v1 v be the      
volume of unit of mass in the first and 
second states respectively1 the         
isothermal expansion of the first part  
of the cycle will take place in         
consequence of the conversion of a mass 
dm from the first state to the second4  
,thus dv1 the change of volume1 is dm7v 
v74 ,also if ,l be the latent heat of   
the substance in the second state1 e4g4 
the latent heat of water1 ,mdv  ,ldm2 so
 that ,m7v  v7  ,l4 ,if dp be the step  
of pressure corresponding to the step   
d,t of temperature1 equation 7,a7       
becomes    d,t,t  dp7v  v7,l            
444 7,b7  ,in the case of coexistence of
 the liquid and solid phases1 this gives
 us the very remarkable result that a   
change of pressure dp will raise or     
lower the temperature of coexistence of 
the two phases1 that is1 the melting    
point of the solid1 by the difference of

 temperature1 d,t1 according as v is    
greater or less than v ,thus a substance
 like water1 which expands in freezing1 
so that v  v is negative1 has its       
freezing point lowered by increase of   
pressure and raised by diminution of    
pressure4 ,this is the result predicted 
by ,professor ,james ,thomson and       
verified experimentally by his brother  
7p4 #aac above74 ,on the other hand1 a  
substance like paraffin wax1 which      
contracts in solidifying1 would have its
 melting point raised by increase of    
pressure and lowered by a diminution of 
pressure4  ,the same conclusions would  
be applicable when the phases are liquid
 and vapour of the same substance1 if   
there were any case in which v  v is    
negative4 ,as it is we see1 what is well
 known to be the case1 that the         
temperature of equilibrium of a liquid  
with its vapour is raised by increase of
 pressure4  ,another important result of
 equation 7,b71 as applied to the liquid
 and vapour phases of a substance1 is   

the information which it gives as to the
 density of the saturated vapour4 ,when 
the two phases coexist the pressure is a
 function of the temperature only4      
,hence if the relation of pressure to   
temperature is known1 dpd,t can be      
calculated1 or obtained graphically from
 a curve2 and the volume v per unit mass
 of the vapour will be given in terms of
 dpd,t1 the temperature ,t1 and the     
volume v per unit mass of the liquid4   
,the density of saturated steam at      
different temperatures is very difficult
 to measure experimentally with any     
approach of accuracy3 but so far as     
experiment goes equation 7,b7 is        
confirmed4 ,the theory here given is    
fully confirmed by other results1 and   
equation 7,b7 is available for the      
calculation of v for any substance for  
which the relation between p and ,t is  
known4 ,it is thus that the density of  
saturated steam can best be found4  ,we 
can obtain another important result for 
the case of the working substance in two

 phases from equation 7,b74 ,the        
relation is    ,l,t  c  h  ,l,t         
444 7,c7  where c and h are the specific
 heats of the substance in the two      
phases respectively1 and ,l is the      
latent heat of the second phase at      
absolute temperature ,t4  ,we shall     
obtain the relation in another way1     
which will illustrate another mode of   
dealing with a cycle of operations which
 ,thomson employed4 ,any small step of  
change of a substance may be regarded as
 made up of a step of volume1 say1      
followed by a step of temperature1 that 
is1 by an isothermal step followed by an
 adiabatic step4 ,in this way any cycle 
of operations whatever may be regarded  
as made up of a series of ,carnot       
cycles4 ,but without regarding any cycle
 of a more general kind than ,carnot's  
as thus compounded1 we can draw         
conclusions from it by the dynamical    
theory provided only it is reversible4  
,suppose a gramme1 say1 of the substance
 to be taken at a specified temperature 

,t in the lower phase1 and to be changed
 to the other phase at that temperature4
 ,the heat taken in will be ,l and the  
expansion will be v  v4 ,next1 keeping  
the substance in the second phase1 and  
in equilibrium with the first phase     
7that is1 for example1 if the second    
phase is saturated vapour1 the          
saturation is to continue in the further
 change71 let the substance be lowered  
in temperature by d,t4 ,the heat given  
out by the substance will be hd,t1 where
 h is the specific heat of the substance
 in the second phase4 ,now at the new   
temperature ,t  d,t let the substance be
 wholly brought back to the second      
phase2 the heat given out will be ,l    
,l,t4d,t4 ,finally1 let the substance1  
now again all in the first phase1 be    
brought to the original temperature3 the
 heat taken in will be cdt1 where c is  
the specific heat in the first phase4   
,thus the net excess of heat taken in   
over heat given out in the cycle is     
7,l,t  c  h7d,t4 ,this must1 in the     

indicator diagram for the changes       
specified1 be the area of the cycle or  
7v  v7p,t4d,t4 ,but by equation 7,b7    
,l,t7v  v7  p,t1 and the area of the    
cycle is 7,l,t7d,t4 ,equating the two   
expressions thus found for the area we  
get equation 7,c74  ,this relation was  
arrived at by ,clausius in his paper    
referred to above1 and the priority of  
publication is his3 it is here given in 
the form which it takes when ,thomson's 
scale of absolute temperature is used4  
,regnault's experimental results for the
 heat required to raise unit mass of    
water from the temperature of melting   
ice to any higher temperature and       
evaporate it at that temperature enable 
the values of ,l,t and ,l,t to be       
calculated1 and therefore that of h to  
be found4 ,it appears that h is negative
 for all the temperatures to which      
,regnault's experimental results can be 
held to apply4 ,this1 as was pointed out
 by ,thomson1 means that if a mass of   
saturated vapour is made to expand so as

 at the same time to fall in            
temperature1 it must have heat given to 
it1 otherwise it will be partly         
condensed into liquid2 and1 on the other
 hand1 if the vapour be compressed and  
made to rise in temperature while at the
 same time it is kept saturated1 heat   
must be taken from it1 otherwise the    
vapour will become superheated and so   
cease to be saturated4  ,it is          
convenient to notice here the article on
 ,heat which ,thomson wrote for the     
ninth edition of the ,encyclopdia       
,britannica4 ,in that article he gave a 
valuable discussion of ordinary         
thermometry1 of thermometry by means of 
the pressures of saturated vapour of    
different substances--steam-pressure    
thermometers1 he called them--of        
absolute thermodynamic thermometry1 all 
enriched with new experimental and      
theoretical investigations1 and appended
 to the whole a valuable synopsis1 with 
additions of his own1 of the ,fourier   
mathematics of heat conduction4  ,first 

dealing with temperature as measured by 
the expansion of a liquid in a less     
expansible vessel1 he showed how it is  
in reality numerically reckoned4 ,this  
amounted to a discussion of the scale of
 an ordinary mercury-in-glass           
thermometer1 a subject concerning which 
erroneous statements are not            
infrequently made in text-books4 ,a     
sketch of ,thomson's treatment of it is 
given here4  ,considering this          
thermometer as a vessel consisting of a 
glass bulb and a long glass stem of fine
 and uniform bore1 hermetically sealed  
and containing only mercury and mercury 
vapour1 he explained the numerical      
relation between the temperature as     
shown by the instrument and the volumes 
of the mercury and vessel4 ,the scale is
 really defined by the method of        
graduation adopted4 ,two points of      
reference are marked on the stem at     
which the top of the mercury stands when
 the vessel is immersed 7#a7 in melting 
ice1 7#b7 in saturated steam under      

standard atmospheric pressure4 ,the stem
 is divided into parts of equal volume  
of bore between these two points and    
beyond each of them4 ,for a centigrade  
thermometer the bore-space between the  
two points is divided into #ajj equal   
parts1 and the lower point of reference 
is marked #j and the upper #ajj1 and the
 other dividing marks are numbered in   
accordance with this along the stem4    
,each of these parts of the bore may be 
called a degree-space4  ,now let the    
instrument contain in its bulb and stem1
 up to the mark #j1 ,n degree-spaces1   
and let v be the volume of a            
degree-space at that temperature4 ,the  
volume up to the mark #j will be ,nv1 at
 that temperature2 and if the substance 
of the vessel be quite uniform in       
quality and free from stress1 ,n will be
 the same for all temperatures4 ,if v be
 the volume of a degree-space at the    
temperature of melting ice the volume of
 the mercury at that temperature will be
 ,nv4 ,if ,g be the expansion of the    

glass when the volume of a degree-space 
is increased from v to v by the rise of 
temperature1 then v  v7#a  ,g74 ,the    
volume of the mercury has been increased
 therefore to 7,n  n7v7#a  ,g7 by the   
same rise of temperature1 if the top of 
the column is thereby made to rise from 
the mark #j so as to occupy n           
degree-spaces more than before4 ,but if 
,e be the expansion of the mercury      
between the temperature of melting ice  
and that which has now been attained1   
the volume of the mercury is also ,nv7#a
  ,e74 ,hence ,n7#a  ,e7  7,n  n77#a    
,g74 ,this gives n  ,n7,e  ,g77#a  ,g74 
,if we take1 as is usual1 n as measuring
 the temperature1 and substitute for it 
the symbol t1 we have1 since ,n  #ajj7a 
,g77,e  ,g71    t  #ajj 7#a  ,g77#a  ,g7
 7,e  ,g77,e  ,g7     444 7,d7  ,in this
 reckoning the definition of any        
temperature1 let us say #cg ,c41 is the 
temperature of the vessel and its       
contents when the top of the mercury    
column stands at the mark #cg above #j1 

on the scale defined by the graduation  
of the instrument2 but the numerical    
signification with relation to the      
volumes is given by equation 7,d74 ,this
 shows that the numerical measure of any
 temperature involves both the expansion
 of the vessel and that of the glass    
vessel between the temperature of       
melting ice and the temperature in      
question4 ,this result may be contrasted
 with the erroneous statement frequently
 made that equal increments of          
temperature correspond to equal         
increments of the volume of the         
thermometric substance4 ,it also shows  
that different mercury-in-glass         
thermometers1 however accurately made   
and graduated1 need not agree when      
placed in a bath at any other           
temperature than #j ,c4 or #ajj ,c4     
,this fact1 and the results of the      
comparison of thermometers made with    
different kinds of glass with the normal
 air thermometer1 which was carried out 
by ,regnault1 were always insisted on by

 ,thomson in his teaching when he dealt 
with the subject of heat4 ,the scale of 
a mercury-in-glass thermometer is too   
often in text-books1 and even in ,acts  
of ,parliament regarded as a perfectly  
definite thing1 and the expansion of a  
gas is not infrequently defined by this 
indefinite scale1 instead of being used 
as it ought to be1 as the basis of      
definition of the scale of the gas      
thermometer4 ,the whole treatment of the
 so-called gaseous laws is too often1   
from a logical point of view1 a mass of 
confusion4  ,in his article on ,heat    
,thomson gave two definitions of the    
scale of absolute temperature4 ,one is  
that stated on p4 #abf above1 namely1   
that the temperature of the source and  
refrigerator are in the ratio of the    
heat taken in from the source to the    
heat given to the refrigerator1 when the
 engine describes a ,carnot cycle       
consisting of two isothermal and two    
adiabatic changes4  ,the other          
definition is better adapted for general

 use1 as it applies to any cycle        
whatever which is reversible4 ,let the  
working substance expand under constant 
pressure by an amount dv 7,,ab' in ,fig4
 #ab71 and let heat ,h be given to the  
substance at the same time4 ,the        
external work done is pdv4 ,thomson     
called pdv,h the work ratio4 ,now let   
the temperature be raised by d,t without
 giving heat to the substance or taking 
heat from it1 and let the corresponding 
pressure rise be dp2 and call dpp the   
pressure ratio4 ,the temperature ratio  
d,t,t is equal to the product of the    
work ratio and the pressure ratio1 that 
is1    d,t,t  dvdp,h  ,this is clearly  
true2 for dvdp is the area of a cycle   
like ,,ab',c',d1 represented in ,fig4   
#ab1 for which an amount of heat ,h is  
taken in1 though not in this case       
strictly at one temperature4 ,and       
clearly1 since in ,fig4 #ab the change  
from ,b' to ,b is adiabatic1 ,h is the  
heat which would have to be taken in for
 the isothermal change ,,ab in the      

,carnot cycle ,,abcd1 which has the same
 area as ,,ab',c',d4 ,thus the          
efficiency of the cycle is dvdp,h1 and  
this by the former definition is d,t,t4 
,or we may regard the matter thus3--,the
 amount of heat ,h which corresponds to 
an infinitesimal expansion dv may be    
used in equation 7,a7 whether the       
expansion is isothermal or not1 if we   
take ,t as the average temperature of   
the expansion4 ,hence we have dpd,t     
,h7dv4,t71 that is1 d,t,t  dpdv,h4 ,the 
theorem on p4 #abh is obtained by what  
is virtually this process4              
,,comparison ,,of ,,absolute ,,scale    
,,with ,,scale ,,of ,,air ,,thermometer 
,the comparison which ,joule and        
,thomson carried out of the absolute    
thermodynamic scale with the scale of   
the constant pressure gas thermometer   
has already been referred to1 and it has
 been shown that the two scales would   
exactly agree1 that is1 absolute        
temperature would be simply proportional
 to the volume of the gas in a gas      

thermometer kept at the temperature to  
be measured1 if the internal energy of  
the gas were not altered by an          
alteration of volume without alteration 
of temperature1 that is1 if the de  e of
 p4 #ajg above were zero4 ,joule tested 
whether this was the case by immersing  
two vessels1 connected by a tube which  
could be opened or closed by a stopcock1
 in the water of a calorimeter1         
ascertaining the temperature with a very
 sensitive thermometer1 and then        
allowing air which had already been     
compressed into one of the vessels to   
flow into the other1 which was initially
 empty4 ,it was found that no alteration
 of temperature of the water of the     
calorimeter that could be observed was  
produced4 ,but the volume of the air had
 been doubled by the process1 and if any
 sensible alteration of internal energy 
had taken place it would have shown     
itself by an elevation or a lowering of 
the temperature of the water1 according 
as the energy had been diminished or    

increased4  ,thomson suggested that the 
gas to be examined should be forced     
through a pipe ending in a fine nozzle1 
or1 preferably1 through a plug of porous
 material placed in a pipe along which  
the gas was forced by a pump1 and       
observations made of the temperature in 
the steady stream on both sides of the  
plug4 ,the experiments were carried out 
with a plug of compressed cotton-wool   
held between two metal disks pierced    
with holes1 in a tube of boxwood        
surrounded also by cotton-wool1 and     
placed in a bath of water closely       
surrounding the supply pipe4 ,this was  
of metal1 and formed the end of a long  
spiral all immersed in the bath4 ,thus  
the temperature of the gas approaching  
the plug was kept at a uniform          
temperature determined by a delicate    
thermometer2 another thermometer gave   
the temperature in the steady stream    
beyond the plug4  ,in the case of       
hydrogen the experiments showed a slight
 heating effect of passage through the  

plug2 air1 oxygen1 nitrogen and carbonic
 acid were cooled by the passage4  ,the 
theory of the matter is set forth in the
 original papers1 and in a very elegant 
manner in the article on ,heat4 ,the    
result of the analysis shows that if w  
be the positive or negative work-value  
of the heat which will convert one      
gramme of the gas after passage to its  
original temperature2 and ,t be absolute
 temperature1 and v volume of a gramme  
of the gas at pressure p1 and the       
difference of pressure on the two sides 
of the plug be dp1 the equation which   
holds is    7#a,t7 7,tv7  #av  7wdp7    
444 7,e7#ah  ,it was found by ,joule and
 ,thomson that w was proportional to dp 
for values of dp up to five or six      
atmospheres4 ,at different temperatures1
 however1 in the case of hydrogen the   
heating effect was found to diminish    
with rise of temperature1 being 4#ajj of
 a degree centigrade at #d or #e        
centigrade1 and 4#aee at temperatures of
 from #hi to #ic centigrade for a       

difference of pressure due to #ajj      
inches of mercury4  ,if there is neither
 heating nor cooling w  #j1 and we      
obtain by integration ,t  ,cv1 where ,c 
is a constant4  ,elaborate discussions  
of the theory of this experiment will be
 found in modern treatises on           
thermodynamics1 and in various recent   
memoirs1 and the differential equation  
has been modified in various ways1 and  
integrated on various suppositions1     
which it would be out of place to       
discuss here4  ,the cooling effect of   
passing a gas such as air or oxygen     
through a narrow orifice has been used  
to liquefy the gas4 ,the stream of gas  
is pumped along a pipe towards the      
opening1 and that which has passed the  
orifice and been slightly cooled is led 
on its way back to the pump along the   
outside of the pipe by which more gas is
 approaching the orifice1 and so cools  
slightly the advancing current4 ,the gas
 which emerges later is thus cooler than
 that which emerged before1 and the     

process goes on until the issuing gas is
 liquefied and falls down into the lower
 part of the pipe surrounding the       
orifice1 whence it can be drawn off into
 vessels constructed to receive and     
preserve it4  ,it is possible thus to   
liquefy hydrogen1 which shows that at   
the low temperature at which the process
 is usually started 7an initial cooling 
is applied7 the passage through the     
orifice has a cooling effect as in the  
other cases4  ,another idea1 that of    
thermodynamic motivity1 on which        
,thomson suggested might be founded a   
fruitful presentation of the subject of 
thermodynamics1 may be mentioned here4  
,it was set forth in a letter written to
 ,professor ,tait in ,may #ahgi4 ,if a  
system of bodies be given1 all at       
different temperatures1 it is possible  
to reduce them to a common temperature1 
and by doing so to extract a certain    
amount of mechanical energy from them4  
,the temperatures must for this purpose 
be equalised by perfect thermodynamic   

engines working between the final       
temperature ,t1 say1 and the            
temperatures of the different parts of  
the system4 ,this process is one of the 
levelling up and the levelling down of  
temperature2 and the temperature ,t is  
such that exactly the heat given out at 
,t by certain engines1 receiving heat   
from bodies of higher temperature than  
,t1 is supplied to the engines which    
work between ,t and bodies at lower     
temperatures4 ,the whole useful work    
obtained in this way was called by      
,thomson the motivity of the system4 ,of
 course equalisation of temperature may 
be obtained by conduction1 and in this  
case the energy which might be utilised 
is lost4 ,with two equal and similar    
bodies at absolute temperatures ,t1 ,t' 
the temperature to which they are       
reduced when their motivity is extracted
 is 7,,tt'74 ,if the temperatures are   
equalised by conduction the resulting   
temperature is higher1 being 7,t  ,t'74 
,thus1 if only the two bodies are       

available for engines to work between1  
the motivity is the measure of the      
energy lost when conduction brings about
 equalisation of temperature4  ,a very  
suggestive paper on the subject was     
published by ,lord ,kelvin in the       
,trans4 ,r4,s4,e41 vol4 #bh1 #ahgg-h4   
,,dissipation ,,of ,,energy  ,in        
connection with the theory of heat must 
be mentioned ,thomson's great           
generalisation1 the theory of the       
dissipation of energy4#ai ,most people  
have some notion of the meaning of the  
physical doctrine of conservation of    
energy1 though in popular discourses it 
is usually misstated4 ,what is meant is 
that in a finite material system1 which 
is isolated in the sense that it is not 
acted on by force from without1 the     
total amount of energy--that is1 energy 
of motion and energy of relative        
position 7including energy of chemical  
affinity7 of the parts--remains         
constant4 ,the usual misstatement is    
that the energy of the universe is      

constant4 ,this may be true if the      
universe is finite2 if the universe is  
infinite in extent the statement has no 
meaning4 ,in any case1 we know nothing  
about the universe as a whole1 and      
therefore make no statements regarding  
it4  ,but while there is thus           
conservation or constancy of amount of  
energy in an isolated and finite        
material system1 this energy may to     
residents on the system become          
unavailable4 ,for useful work within    
such a system is done by conversion of  
energy from one form to another and the 
total amount remains unchanged4 ,but if 
this conversion is prevented all        
processes which involve such conversion 
must cease1 and among these are vital   
processes4  ,the unavailable form which 
the energy of the system with which we  
are directly and at present concerned1  
whatever may become of us ultimately1 is
 taking1 according to ,thomson's theory1
 is universally diffused heat4 ,how this
 comes about may be seen as follows4    

,even a perfect engine1 if the          
refrigerator be at the lowest available 
temperature1 rejects a quantity of heat 
which cannot be utilised for the        
performance of the work4 ,this heat is  
diffused by conduction and radiation to 
surrounding bodies1 and so to bodies    
more remote1 and the general temperature
 of the system is raised4 ,moreover1 as 
heat engines are imperfect there is heat
 rejected to the surroundings by        
conduction1 and produced by work done   
against friction1 so that the heat      
thrown on the unavailable or waste heap 
is still further increased4  ,conduction
 of heat is the great agency by which   
energy is more and more dispersed in    
this unavailable form throughout the    
totality of material bodies4 ,as has    
been seen1 available motivity is        
continually wasted through its agency2  
and in the flow of heat in the earth and
 in the sun and other unequally heated  
bodies of our system the waste of energy
 is prodigious4 ,aided by convection    

currents in the air and in the ocean it 
continually equalises temperatures1 but 
does so at an immense cost of useful    
energy4  ,then in our insanely wasteful 
methods of heating our houses by open   
fires1 of half burning the coal used in 
boiler furnaces1 and allowing unconsumed
 carbon to escape into the atmosphere in
 enormous quantities1 while a very large
 portion of the heat actually generated 
is allowed to escape up chimneys with   
heated gases1 the store of unavailable  
heat is being added to at a rate which  
will entail great distress1 if not ruin1
 on humanity at no indefinitely distant 
future4 ,it will be the height of       
imprudence to trust to the prospect1 not
 infrequently referred to at the present
 time1 of drawing on the energy locked  
up in the atomic structure of matter4   
,he would be a foolish man who would    
wastefully squander the wealth he       
possesses1 in the belief that he can    
recoup himself from mines which all     
experience so far shows require an      

expenditure to work them far beyond any 
return that has as yet been obtained4   
,it is not apart from our present theme 
to urge that it is high time the        
question of the national economy of     
fuel1 and the desirability of utilising 
by afforestation the solar energy       
continually going to waste on the       
surface of the earth1 were dealt with by
 statesmen4 ,if statesmen would but make
 themselves acquainted with the results 
of physical science in this magnificent 
region of cosmic economics there would  
be some hope1 but1 alas6 as a rule their
 education is one which inevitably leads
 to neglect1 if not to disdain of       
physical teaching4  ,from the causes    
which have been referred to1 energy is  
continually being dissipated1 not       
destroyed1 but locked up in greater and 
greater quantity in the general heat of 
bodies4 ,there is always friction1      
always heat conduction and convection1  
so that as our stores of motional or    
positional energy1 whether of chemical  

substances uncombined1 the earth's      
motion1 or what not1 are drawn upon1 the
 inevitable fraction1 too often a large 
proportion1 is shed off and the general 
temperature raised4 ,after a large part 
of the whole existent energy has gone   
thus to raise the dead level of things1 
no difference of temperature adequate   
for heat engines to work between will be
 possible1 and the inevitable death of  
all things will approach with headlong  
rapidity4   ,,thermoelasticity ,,and    
,,thermoelectricity  ,in the second     
definition of the scale of absolute     
temperature just discussed1 stress of   
any type may be substituted for         
pressure1 and the corresponding         
displacement s for the change of volume4
 ,thus for a piece of elastic material  
put through a cycle of changes we may   
substitute d,s for dp and ,ads for dv2  
where ,a is such a factor that ,ad,sds  
is the work done in the displacement ds 
by the stress d,s4 ,as an example       
consider a wire subjected to simple     

longitudinal stress ,s4 ,longitudinal   
extension is produced1 but this is not  
the only change2 there is at the same   
time lateral contraction4 ,however1 s   
within certain limits is proportional to
 ,s4  ,let heat d,h in dynamical measure
 be given to the wire while the stress  
,s is maintained constant1 and let the  
extension increase from s to s  ds4 ,the
 stress ,s will do work ,a,sds on the   
wire1 and the work ratio will be        
-,a,sdsd,h4 ,now let the stress be      
increased to ,s  d,s while the extension
 is kept constant1 and the absolute     
temperature raised from ,t to ,t  d,t4  
,the stress ratio 7as we may call it7 is
 d,s,s and the temperature ratio d,t,t4 
,thus we obtain 7p4 #acd above7         
7d,sd,t7  7#a,,ta7 7d,hds7  ,in his     
,heat article ,thomson used the         
alteration e of strain under constant   
stress 7that is dsl1 where l is the     
length of the wire7 corresponding to an 
amount of heat sufficient to raise the  
temperature under constant stress by #a4

 ,hence if ,k be the specific heat under
 constant stress1 and le be put for ds  
in the sense just stated1 we have    d,t
  7,ted,s,k7                            
444 7,f7  where  is the density1 since  
d,h  ,kl,a4  ,the ratio of d,h to the   
increase ds of the extension is positive
 or negative1 that is1 the substance    
absorbs or evolves heat1 when strained  
under the condition of constant stress1 
according as d,sd,t is negative or      
positive4 ,or we may put the same thing 
in another way which is frequently      
useful4 ,if a wire subjected to constant
 stress has heat given to it1 ds is     
negative or positive1 in other words the
 wire shortens or lengthens1 according  
as d,sd,t is positive or negative1 that 
is1 according as the stress for a given 
strain is increased or diminished by    
increase of temperature4  ,it is known  
from experiment that a metal wire       
expands under constant stress when heat 
is given to it1 and thus we learn from  
the equation 7,f7 that the stress       

required for a given strain is          
diminished when the temperature of the  
wire is raised4 ,again1 a strip of      
india-rubber stretched by a weight is   
shortened if its temperature is raised1 
consequently the stress required for a  
given strain is increased by rise of    
temperature4  ,these results1 from a    
qualitative point of view1 are          
self-evident4 ,but from what has been   
set forth it will be obvious that an    
equation exactly similar to 7,f7 holds  
whether the change ds of s is taken as  
before under constant stress1 or at     
uniform temperature1 or whether the     
change d,s of ,s is effected            
adiabatically or at constant strain4    
,in all these cases the same equation   
d,t  ,t 7ed,s,k7                        
444 7,g7  applies1 with the change of   
meaning of d,t involved4  ,this equation
 differs from that of ,thomson as given 
in various places 7e4g4 in the          
,encyclopdia ,britannica article on     
,elasticity which he also wrote7 in the 

negative sign on the right-hand side1   
but the difference is only apparent4    
,according to his specification a       
pressure would be a positive stress1 and
 an expansion a positive displacement1  
and in applying the equation to         
numerical examples this must be borne in
 mind so that the proper signs may be   
given to each numerical magnitude4 ,as  
an example of adiabatic change1 a sudden
 extension of the wire already referred 
to by an increase of stress d,s may be  
considered4 ,if there is not time for   
the passage of heat from or to the      
surroundings of the wire1 the change of 
temperature will be given by equation   
7,g74  ,this equation was applied by    
,thomson 7article ,elasticity7 to find  
the relation between what he called the 
kinetic modulus of elasticity and the   
static modulus1 that is1 between the    
modulus for adiabatic strain and the    
modulus for isothermal strain4  ,the    
augmentation of the strain produced by  
raising the temperature #a is e1 and    

therefore ed,t1 that is1 ,ted,s,k1 is   
the increase of strain due to the sudden
 rise of temperature d,t4 ,this added to
 the isothermal strain produced by d,s  
will give the whole adiabatic strain4   
,thus if ,m be the static or isothermal 
modulus1 the adiabatic strain is d,s,m  
,ted,s,k4 ,if ,m' denote the kinetic or 
adiabatic modulus its value is d,s      
divided by the whole adiabatic strain1  
that is1 ,m'  ,m7#a  ,m,te,k7 and the   
ratio ,m',m  #a7a  ,m,te,k74  ,it is    
well known and easy to prove1 without   
the use of any theorem which can be     
properly called thermodynamic1 that this
 ratio of moduli is equal to the ratio  
of the specific heat ,k of the          
substance1 under the condition of       
constant stress1 to the specific heat ,n
 under the condition of constant strain 
of the corresponding type4 ,this1       
indeed1 is self-evident if two changes  
of stress1 one isothermal the other     
adiabatic1 which produce the same steps 
of displacement ds1 be considered1 and  

it be remembered that the step ,t of    
temperature which accompanies the       
adiabatic change may be regarded as made
 up of a step d,t of temperature1       
accompanying a displacement ds effected 
at constant stress1 and then two        
successive steps d,t and ,t effected1 at
 constant strain1 along with the steps  
of stress d,s4 ,the ratio ,m',m is      
easily seen to have the value 7,t       
d,t7dt1 and since ,kd,t  ,n7,t  d,t7    
#j1 by the adiabatic condition1 the     
theorem is proved4  ,laplace's          
celebrated result for air1 according to 
which the adiabatic bulk-modulus is     
equal to the static bulk-modulus        
multiplied by the ratio of the specific 
heat of air pressure constant to the    
specific heat of air volume constant1 is
 a particular example of this theory4   
,thomson showed in the ,elasticity      
article how1 by the value of ,m',m1     
derived as above from thermodynamic     
theory1 the value of ,k,n could be      
obtained for different substances and   

for different types of stress1 and gave 
very interesting tables of results for  
solids1 liquids1 and gases subjected to 
pressure-stress 7bulk-modulus7 and for  
solids subjected to longitudinal stress 
7,young's modulus74  ,the discussion as 
to the relation of the adiabatic and    
isothermal moduli of elasticity is part 
of a very important paper on            
,thermoelastic1 ,thermomagnetic1 and    
,thermoelectric ,properties of ,matter1 
which he published in the ,philosophical
 ,magazine for ,january #ahgh4 ,this was
 in the main a reprint of an article    
entitled1 ,on the ,thermoelastic and    
,thermomagnetic ,properties of ,matter1 
,part ,i1 which appeared in ,april #ahee
 in the first number of the ,quarterly  
,journal of ,mathematics4 ,only         
thermoelasticity was considered in this 
article2 the thermomagnetic results had1
 however1 been indicated in an article  
on ,thermomagnetism in the second       
edition of the ,cyclopdia of ,physical  
,science1 edited and in great part      

written by ,professor ,j4 ,p4 ,nichol1  
and published in #ahfj4 ,for the same   
,cyclopdia ,thomson also wrote an       
article entitled1 ,thermo-electric1     
,division ,i4--,pyro-,electricity1 or   
,thermo-,electricity of ,non-conducting 
,crystals1 and the enlarged ,phil4 ,mag4
 article also contained the application 
of thermodynamics to this kind of       
thermoelectric action4  ,this great     
paper cannot be described without a good
 deal of mathematical analysis2 but the 
student who has read the earlier        
thermodynamical papers of ,thomson will 
have little difficulty in mastering it4 
,it must suffice to say here that it may
 be regarded as giving the keynote of   
much of the general thermodynamic       
treatment of physical phenomena1 which  
forms so large a part of the physical   
mathematics of the present day1 and     
which we owe to ,willard ,gibbs ,duhem1 
and other contemporary writers4         
,thomson had1 however1 previous to the  
publication of this paper1 applied      

thermodynamic theory to thermoelectric  
phenomena4 ,a long series of papers     
containing experimental investigations1 
and entitled1 ,electrodynamic ,qualities
 of ,metals1 are placed in the second   
volume of his ,mathematical and         
,physical ,papers4 ,this series begins  
with the ,bakerian ,lecture 7published  
in the ,transactions of the ,royal      
,society for #ahef7 which includes an   
account of the remarkable experimental  
work accomplished during the preceding  
four or five years by the volunteer     
laboratory corps in the                 
newly-established physical laboratory in
 the old ,college4 ,the subjects dealt  
with are the ,electric ,convection of   
,heat1 ,thermoelectric ,inversions1 the 
,effects of ,mechanical ,strain and of  
,magnetisation on the ,thermoelectric   
,qualities of ,metals1 and the ,effects 
of ,tension and ,magnetisation on the   
,electric ,conductivity of ,metals4 ,it 
is only possible to give here a very    
short indication of the thermodynamic   

treatment1 and of the nature of         
,thomson's remarkable discovery of the  
electric convection of heat4  ,it was   
found by ,seebeck in #ahbb that when a  
circuit is formed of two different      
metals 7without any cell or battery7 a  
current flows round the circuit if the  
two junctions are not at the same       
temperature4 ,for example1 if the two   
metals be rods of antimony and bismuth1 
joined at their extremities so as to    
form a complete circuit1 and one        
junction be warmed while the other is   
kept at the ordinary temperature1 a     
current flows across the hot junction in
 the direction from bismuth to antimony4
 ,similarly1 if a circuit be made of a  
copper wire and an iron wire1 a current 
passes across the warmer junction from  
copper to iron4 ,the current            
strength--other things being the        
same--depends on the metals used2 for   
example1 bismuth and antimony are more  
effective than other metals4  ,it was   
found by ,peltier that when a current1  

say from a battery1 is sent round such a
 circuit1 that junction is cooled and   
that junction is heated by the passage  
of the current1 which1 being            
respectively heated and cooled1 would   
without the cell have caused a current  
to flow in the same direction4 ,thus the
 current produced by the difference of  
temperature of the junctions causes an  
absorption of heat from the warmer      
junction1 and an evolution of heat at   
the colder junction4  ,this naturally   
suggested to ,thomson the consideration 
of a circuit of two metals1 with the    
junctions at different temperatures1 as 
a heat engine1 of which the hot junction
 was the source and the cold junction   
the refrigerator1 while the heat        
generated in the circuit by the current 
and other work performed1 if there was  
any1 was the equivalent of the          
difference between the heat absorbed and
 the heat evolved4 ,of course in such an
 arrangement there is always            
irreversible loss of heat by conduction2

 but when such losses are properly      
allowed for the circuit is capable of   
being correctly regarded as a reversible
 engine4  ,shortly after ,seebeck's     
discovery it was found by ,cumming that 
when the hot junction was increased in  
temperature the electromotive force     
increased more and more slowly1 at a    
certain temperature of the hot junction 
took its maximum value1 and then as the 
temperature of the hot junction was     
further increased began to diminish1 and
 ultimately1 at a sufficiently high     
temperature1 in most instances changed  
sign4 ,the temperature of maximum       
electromotive force was found to be     
independent of the temperature of the   
colder junction4 ,it is called the      
temperature of the neutral point1 from  
the fact that if the two junctions of a 
thermoelectric circuit be kept at a     
constant small difference of            
temperature1 and be both raised in      
temperature until one is at a higher    
temperature than the neutral point1 and 

the other is at a lower1 the            
electromotive force will fall off1 until
 finally1 when this point is reached1 it
 has become zero4  ,thus it was found   
that for every pair of metals there was 
at least one such temperature of the hot
 junction1 and it was assumed1 with     
consequences in agreement with          
experimental results1 that when the     
temperature was the neutral temperature 
there was neither absorption nor        
evolution of heat at the junction4 ,but 
then the source provided by the         
thermodynamic view just stated had      
ceased to exist4 ,the current still     
flowed1 there was evolution of heat at  
the cold junction1 and likewise ,joulean
 evolution of heat in the wires of the  
circuit in consequence of their         
resistance4 ,hence it was clear that    
energy must be obtained elsewhere than  
at the junctions4 ,thomson solved the   
problem by showing that 7besides the    
,joulean evolution of heat7 there is    
absorption 7or evolution7 of heat when a

 current flows in a conductor along     
which there is a gradient of            
temperature4 ,for example1 when an      
electric current flows along an         
unequally heated copper wire1 heat is   
evolved where the current flows from the
 hot parts to the cold1 and heat is     
absorbed where the flow is from cold to 
hot4 ,when the hot junction is at the   
temperature of zero absorption or       
evolution of heat--the so-called neutral
 temperature--the heat absorbed in the  
flow of the circuit along the unequally 
heated conductors is greater than that  
evolved on the whole1 by an amount which
 is the equivalent of the energy        
electrically expended in the circuit in 
the same time4  ,it was found1 moreover1
 that the amount of heat absorbed by a  
given current in ascending or descending
 through a given difference of          
temperature is different in different   
metals4 ,when the current was unit      
current and the temperature difference  
also unity1 ,thomson called the heat    

absorbed or evolved in a metal the      
specific heat of electricity in the     
metal1 a name which is convenient in    
some ways1 but misleading in others4    
,the term rather conveys the notion that
 electricity has a material existence4  
,a substance such as copper1 lead1      
water1 or mercury has a specific heat in
 a perfectly understood sense2          
electricity is not a substance1 hence   
there cannot be in the same proper sense
 a specific heat of electricity4        
,however1 this absorption and evolution 
of heat was investigated experimentally 
and mathematically by ,thomson1 and is  
generally now referred to in            
thermoelectric discussions as the       
,thomson effect4  ,part ,,vi 7,trans4   
,r4,s41 #ahge7 of the investigations of 
the electrodynamic qualities of metals  
dealt with the effects of stretching and
 compressing force1 and of torsion1 on  
the magnetisation of iron and steel and 
of nickel and cobalt4  ,one of the      
principal results was the discovery that

 the effect of longitudinal pull is to  
increase the inductive magnetisation of 
soft iron1 and of transverse thrust to  
diminish it1 so long as the magnetising 
field does not exceed a certain value4  
,when this value1 which depends on the  
specimen1 is exceeded1 the effect of    
stress is reversed4 ,the field-intensity
 at which the effect is reversed is     
called the ,villari critical intensity1 
from the fact1 afterwards ascertained1  
that the result had previously been     
established by ,villari in ,italy4 ,no  
such critical value of the field was    
found to exist for steel1 or nickel1 or 
cobalt4  ,in some of the experiments the
 specimen was put through a cycle of    
magnetic changes1 and the results       
recorded by curves4 ,these proved that  
in going from one state to another and  
returning the material lagged in its    
return path behind the corresponding    
states in the outward path4 ,this is the
 phenomenon called later hysteresis1 and
 studied in minute detail by ,ewing and 

others4 ,thomson's magnetic work was    
thus the starting point of many more    
recent researches4     ,,chapter ,,ix   
,,hydrodynamics--,,dynamical ,,theorem  
,,of ,,minimum ,,energy--,,vortex       
,,motion   ,thomson devoted great       
attention from time to time to the      
science of hydrodynamics4 ,this is      
perhaps the most abstruse subject in the
 domain of applied mathematics1 and when
 viscosity 7the frictional resistance to
 the relative motion of particles of the
 fluid7 is taken into account1 passes   
beyond the resources of mathematical    
science in its present state of         
development4 ,but leaving viscosity     
entirely aside1 and dealing only with   
so-called perfect fluids1 the           
difficulties are often overwhelming4    
,for a long time the only kind of fluid 
motion considered was1 with the         
exception of a few simple cases1 that   
which is called irrotational motion4    
,this motion is characterised by the    
analytical peculiarity1 that the        

velocity of an element of the fluid in  
any direction is the rate of variation  
per unit distance in that direction of a
 function of the coordinates 7the       
distances which specify the position7 of
 the particle4 ,this condition very much
 simplifies the analysis2 but when it   
does not hold we have much more serious 
difficulties to overcome4 ,then the     
elements of the fluid have what is      
generally1 but quite improperly1 called 
molecular rotation4 ,for we know little 
of the molecules of a fluid2 even when  
we deal with infinitesimal elements1 in 
the analysis of fluid motion1 we are    
considering the fluid in mass4 ,but what
 is meant is elemental rotation1 a      
rotation of the infinitesimal elements  
as they move4 ,we have an example of    
such motion in the air when a ring of   
smoke escapes from the funnel of a      
locomotive or the lips of a             
tobacco-smoker1 in the motion of part of
 the liquid when a cup of tea is stirred
 by drawing the spoon from one side to  

the other1 or when the blade of an oar  
is moving through the water4 ,in these  
last two cases the depressions seen in  
the surface are the ends of a vortex    
which extends between them and          
terminates on the surface4 ,in all these
 examples what have been called vortices
 are formed1 and hence the name vortex  
motion has been given to all those cases
 in which the condition of              
irrotationality is not satisfied4  ,the 
first great paper on vortex motion was  
published by von ,helmholtz in #aheh1   
and ten years later a memoir on the same
 subject by ,thomson was published in   
the ,transactions of the ,royal ,society
 of ,edinburgh4 ,in that memoir are     
given very much simpler proofs of von   
,helmholtz's main theorems1 and1        
moreover1 some new theorems of wide     
application to the motion of fluids4    
,one of these is so comprehensive that  
it may be said with truth to contain the
 whole of the dynamics of a perfect     
fluid4 ,we go on to indicate the        

contents of the principal papers1 as far
 as that can be done without the        
introduction of analysis of a difficult 
description4  ,in ,chapter ,,vi         
reference has been made to the ,notes on
 ,hydrodynamics published by ,thomson in
 the ,cambridge and ,dublin             
,mathematical ,journal for #ahdh and    
#ahdi4 ,these ,notes were not intended  
to be entirely original1 but were       
composed for the use of students1 like  
,airy's ,tracts of fifteen years before4
  ,the first ,note dealt with the       
equation of continuity1 that is to say1 
the mathematical expression of the      
obvious fact that if any region of space
 in a moving fluid be considered1 the   
excess of rate of flow into the space   
across the bounding surface1 above the  
rate of flow out1 is equal to the rate  
of growth of the quantity of fluid      
within the space4 ,the proof given is   
that now usually repeated in text-books 
of hydrodynamics4  ,the second ,note    
discussed the condition fulfilled at the

 bounding surface of a moving fluid4    
,the chief mathematical result is the   
equation which expresses the fact1 also 
obvious without analysis1 that there is 
no flow of the fluid across the surface4
 ,in other words1 the component of the  
motion of a fluid particle in the       
immediate neighbourhood of the surface  
at any instant1 taken in the direction  
perpendicular to the surface1 must be   
equal to the motion of the surface in   
that direction at the same instant4     
,the third ,note1 published a year later
 7,february #ahdi71 is of considerable  
scientific importance4 ,it is entitled1 
,on the ,vis ,viva of a ,liquid in      
,motion4 ,what used to be called the vis
 viva of a body is double what is now   
called the energy of motion1 or kinetic 
energy1 of the body4 ,the term liquid is
 merely a brief expression for a fluid1 
the mass of which per unit volume is the
 same throughout1 and suffers no        
variation4 ,the fluid1 moreover1 is     
supposed devoid of friction1 that is1   

the relative motions of its parts are   
unresisted by tangential force between  
them4 ,the chief theorem proved and     
discussed may be described as follows4  
,the liquid is supposed to fill the     
space within a closed envelope1 which   
fulfils the condition of being simply   
continuous4 ,the condition will be      
understood by imagining any two points  
,a1 ,b1 within the space1 to be joined  
by two lines ,,acb1 ,,adb both lying    
within the space4 ,these two lines will 
form a circuit ,,acbda4 ,if now this    
circuit1 however it may be drawn1 can be
 contracted down to a point1 without any
 part of the circuit passing out of the 
space1 the condition is fulfilled4      
,clearly the space within the surface of
 an anchor-ring1 or a curtain-ring1     
would not fulfil this condition1 for one
 part of the circuit might pass from ,a 
to ,b round the ring one way1 and the   
other from ,a to ,b the other way4 ,the 
circuit could not then be contracted    
towards a point without passing out of  

the ring4  ,now let the liquid given at 
rest in such a space be set in motion by
 any arbitrarily specified variation of 
position of the envelope4 ,the liquid   
within will be set in motion in a manner
 depending entirely on the motion of the
 envelope4 ,it is possible to conceive  
of other motions of the liquid than that
 taken1 which all agree in having the   
specified motion of the surface4        
,thomson's theorem asserts that the     
motion actually taken has less kinetic  
energy than that of any of the other    
motions which have the same motion of   
the bounding surface4  ,the motion      
produced has the property described by  
the word irrotational1 that is1 the     
elements of the fluid have no spinning  
motion--they move without rotation4 ,a  
small portion of a fluid may describe   
any path--may go round in a circle1 for 
example--and yet have no rotation4 ,the 
reader may imagine a ball carried round 
in a circle1 but in such a way that no  
line in the body ever changes its       

direction4 ,the body has translation1   
but no spin4  ,irrotationality of a     
fluid is secured1 as stated above1 when 
the velocity of each element in any     
direction is the rate of variation per  
unit distance in that direction of a    
certain function of the coordinates1 the
 distances1 taken parallel to three     
lines perpendicular to one another and  
drawn from a point1 which specify the   
position of the particle4 ,in fact1 what
 is called a velocity-potential exists1 
similar to the potential described in   
,chapter ,,iv above1 for an electric    
field4 ,this condition1 together with   
the specified motion of the surface1    
suffices to determine the motion of the 
fluid4  ,two important particular       
consequences were pointed out by        
,thomson3 7#a7 that the motion of the   
fluid at any instant depends solely on  
the form and motion of the bounding     
surface1 and is therefore independent of
 the previous motion2 and 7#b7 that if  
the bounding surface be instantaneously 

brought to rest1 the liquid throughout  
the vessel will also be instantly       
brought to rest4  ,this theorem was     
afterwards generalised by ,thomson      
7,proc4 ,r4,s4,e41 #ahfc71 and applied  
to any material system of connected     
particles set into motion by specified  
velocities simultaneously and suddenly  
imposed at selected points of the       
system4 ,it was already known that the  
kinetic energy of a system of bodies    
connected in any manner1 and set in     
motion by impulses applied at specified 
points1 was either a maximum or a       
minimum1 as compared with that for any  
other motion compatible with these      
impulses1 and with the connections of   
the system4 ,this was proved by         
,lagrange in the ,mcanique ,analytique  
as a generalisation of a theorem given  
by ,euler for a rigid body set into     
rotation by an impulse4  ,bertrand      
proved in #ahdb that when the impulses  
applied are given in amount1 and are    
applied at specified points1 the system 

starts off with kinetic energy greater  
than that of any other motion which is  
consistent with the given impulses and  
the connections of the system4 ,this    
other motion must be such as could be   
produced in the system by the given     
impulses1 together with any other set of
 impulses capable of doing no work on   
the whole4  ,thomson's theorem is       
curiously complementary to ,bertrand's4 
,let the system be acted on by impulses 
applied at certain specified points1 and
 by no other impulses of any kind2 and  
let the impulses be such as to start    
those selected points with any          
prescribed velocities4 ,the system will 
start off with kinetic energy which is  
less than that of any other motion which
 the system could have consistently with
 the prescribed velocities1 and which it
 could be constrained to take by        
impulses which do no work on the whole4 
,in each case the difference of energies
 is the energy of the motion which must 
be compounded with one motion to give   

the other which is compared with it4  ,a
 simple example1 such as might be taken 
of the particular case considered by    
,euler1 may help to make these theorems 
clear4 ,imagine a straight uniform rod  
to lie on a horizontal table1 between   
which and the rod there is no friction4 
,let the rod be struck a blow at one end
 in a horizontal direction at right     
angles to the length of the rod4 ,if no 
other impulse acts1 the end of the rod  
will move off with a certain definite   
velocity1 and the other parts of the rod
 7which is supposed perfectly unbending7
 will be started by the connections of  
the system4 ,it is obvious that any     
number of other motions of the rod can  
be imagined1 all of which give the same 
motion of the extremity struck4 ,but the
 actual motion taken is one of turning  
about that point of the rod which is    
two-thirds of the length from the end   
struck4 ,if the reader will consider the
 kinetic energy for any other horizontal
 turning motion consistent with the same

 motion of the end1 he will find that   
the kinetic energy is greater than that 
of the motion just specified4 ,this     
motion could be produced by applying at 
the point about which the rod turns the 
impulse required to keep that point at  
rest4 ,the impulse so applied would do  
no work4 ,the actual value is #ahmv1    
where m denotes the mass of the rod and 
v the velocity of the end4 ,if the      
motion taken were one of rotation about 
a point of the rod at distance x from   
the end struck1 the kinetic energy would
 be m7#dl  #flx  #cx7v#fx1 where #bl is 
the length of the rod1 and this has its 
least value #ahmv for x  #dl#c4 ,for    
example1 x  #bl gives #afmv1 which is   
greater than the value just found4      
,bertrand's theorem applied to this case
 of motion is not quite so easy1        
perhaps1 to understand4 ,the motion     
which is said to have maximum energy is 
one given by a specified impulse at the 
end struck1 and this1 in the absence of 
any other impulses1 would be a motion of

 minimum energy4 ,but let the           
alternative motion1 which is to be      
compared with that actually taken1 be   
one constrained by additional impulses  
such as can together effect no work1 and
 the existence of the maximum is        
accounted for4 ,the kinetic energy      
produced is one-half the product of the 
impulse into the velocity of the point  
struck1 that is ,iv1 and it has just    
been seen that this is the product of   
7#af7mv by the factor 7#dl  #flx  #cx7x4
 ,this factor is #c;,imv1 and is a      
minimum when x  #dl#c4 ,thus for a given
 ,i1 v will have its maximum value when 
the factor referred to is least1 and ,iv
 will then be a maximum4  ,the bar can  
be constrained to turn about another    
point by a fixed pivot there situated4  
,an impulse will be applied to the rod  
by the pivot1 simultaneously with the   
blow2 and it is obvious that this       
impulse does no work1 since there is no 
displacement of the point to which it is
 applied4  ,the two theorems are        

consequences of one principle4 ,the     
constraint in each case increases what  
may be called the effective inertia1    
which may be taken as ,iv4 ,thus when v 
is given1 ,i is increased by any        
constraint compelling the rod to rotate 
about a particular axis1 and so ,iv1 or 
the kinetic energy1 is increased4 ,on   
the other hand1 when ,i is given the    
same constraint diminishes v1 and so ,iv
 is diminished4  ,a short paper         
published in the ,b4 ,a4 ,report for    
#aheb points out that the lines of force
 near a small magnet1 placed with its   
axis along the lines of force in a      
uniform magnetic field1 as it would rest
 under the action of the field1 are at  
corresponding points similar to those of
 the field of an insulated spherical    
conductor1 under the inductive influence
 of a distant electric change4 ,further1
 the fact is noted that1 if the magnet  
be oppositely directed to the field1 the
 lines of force are curved outwards1    
just as the lines of flow of a uniform  

stream would be by a spherical obstacle1
 at the surface of which no eddies were 
caused4 ,this is one of those           
instructive analogies between the theory
 of fluid motion and other theories     
involving perfectly analogous           
fundamental ideas1 which ,thomson was   
fond of pointing out1 and which helped  
him in his repeated attempts to imagine 
mechanical representations of physical  
phenomena of different kinds4  ,with    
these may be placed another1 which in   
lectures he frequently dwelt on--a      
simple doublet1 as it is called1        
consisting of a point-source of fluid   
and an equal and closely adjacent       
point-sink4 ,a short tube in an infinite
 mass of liquid1 which is continually   
flowing in at one end and out at the    
other1 may serve as a realisation of    
this arrangement4 ,the lines of flow    
outside the tube are exactly analogous  
to the lines of force of a small magnet2
 and if at the same time there exist a  
uniform flow of the liquid in the       

direction of the length of the tube1 the
 field of flow will be an exact picture 
of the field of force of the small      
magnet1 when it is placed with its      
length along the lines of a previously  
existing uniform field4 ,the flow in the
 doublet will be with or against the    
general flow according as the magnet is 
directed with or against the field4     
,the paper on vortex-motion has been    
referred to above1 and an indication    
given of the nature of the fluid-motion 
described by this title4 ,there are1    
however1 two cases of fluid-motion which
 are referred to as vortices1 though the
 fundamental criterion of               
vortex-motion--the non-existence of a   
velocity-potential--is satisfied in only
 one of them4 ,the exhibition of one of 
these was a favourite experiment in     
,thomson's ordinary lectures1 as his old
 students will remember4 ,if water in a 
large bowl is stirred rapidly with a    
teaspoon carried round and round in a   
circle about the axis of the bowl1 the  

surface will become concave1 and the    
form of the central part will be a      
paraboloid of revolution about the      
vertical through the lowest point1 that 
is to say1 any section of that part of  
the surface made by a vertical plane    
containing the axis will be a parabola  
symmetrical about the axis4 ,the motion 
can be better produced by mounting the  
vessel on a whirling-table1 and rotating
 it about the vertical axis coinciding  
with its axis of figure2 but the        
phenomenon can be quite well seen       
without this machinery4 ,in this case   
the velocity of each particle of the    
water is proportional to its distance   
from the axis1 and the whole mass1 when 
relative equilibrium is set up1 turns1  
as if it were rigid1 about the axis of  
the vessel4 ,each element of the fluid  
in this forced vortex1 as it is called1 
is in rotation1 and1 like the moon1     
makes one turn in one revolution about  
the centre of its path4 ,this is1       
therefore1 a true1 though very simple1  

case of vortex-motion4  ,on the other   
hand1 what may be called a free vortex  
may exist1 and is approximated to       
sometimes when water in a vessel is     
allowed to run off through an escape    
pipe at the bottom4 ,the velocity of an 
element in this vortex is inversely     
proportional to its distance from the   
centre1 and the form of the free surface
 is quite different from that in the    
other case4 ,the name free vortex is    
often given to this case of motion1 but 
there is no vortex-motion about it      
whatever4  ,thomson's great paper on    
vortex-motion was read before the ,royal
 ,society of ,edinburgh in #ahfg1 and   
was recast and augmented in the         
following year4 ,it will be possible to 
give here only a sketch of its scope and
 main results4  ,the fluid is supposed  
contained in a closed fixed vessel which
 is either simply or multiply continuous
 7see p4 #aef71 and may contain immersed
 in it simply or multiply continuous    
solids4 ,when these solids exist their  

surfaces are part of the boundary of the
 liquid2 they are surrounded by the     
liquid unless they are anywhere in      
contact with the containing vessel1 and 
their density is supposed to be the same
 as that of the liquid4 ,they may be    
acted on by forces from without1 and    
they act on the liquid with             
pressure-forces1 and either directly or 
through the liquid on one another4  ,the
 first result obtained is fairly        
obvious4 ,the centre of mass of the     
whole system must remain at rest        
whatever external forces act on the     
solids1 since the density is the same   
everywhere within the vessel1 and the   
vessel is fixed2 that is to say1 there  
is no momentum of the contents of the   
vessel in any direction4 ,for whatever  
motion of the solids is set up by the   
external forces1 must be accompanied by 
a motion of the liquid1 equal and       
opposite in the sense here indicated4   
,after a discussion of what he calls the
 impulse of the motion1 which is the    

system of impulsive forces on the       
movable solids which would generate the 
motion from rest1 ,thomson proceeds to  
prove the important proposition that the
 rotational motion of every portion of  
the liquid mass1 if it is zero at any   
one instant for every portion of the    
mass1 remains always zero4 ,this is done
 by considering the angular momentum of 
any small spherical portion of the      
liquid relatively to an axis through the
 centre of the sphere1 and proving that 
in order that it may vanish1 for every  
axis1 the component velocities of the   
fluid at the centre must be derivable   
from a velocity-potential4 ,the angular 
momentum of a particle about an axis is 
the product of the component of the     
particle's momentum1 at right angles to 
the plane through the particle and the  
axis1 by the distance of the particle   
from the axis4 ,the sum of all such     
products for the particles making up the
 body 7when proper account is taken of  
the signs according to the direction of 

turning round the axis7 is the angular  
momentum4 ,the proof of this result     
adopted is due to ,stokes4 ,the angular 
velocities of an element of fluid at a  
point x1 y1 z1 about the axes of x1 y1 z
 are shown to be 7wy  vz71 etc4  ,the   
condition was therefore shown to be     
necessary2 it remained to prove that it 
was sufficient4 ,this is obvious at once
 from the definition of the             
velocity-potential1 which must now be   
supposed to exist in order that its     
sufficiency may be proved4 ,if any      
diameter of the spherical portion be    
taken as the axis1 and any plane through
 that axis be considered1 the velocity  
of a particle at right angles to that   
plane can be at once expressed as the   
rate at which the velocity-potential    
varies per unit distance along the      
circle1 symmetrical about the axis1 on  
which the particle lies4 ,the integral  
of the velocity-potential round this    
circle vanishes1 and so the angular     
momentum for any thin uniform ring of   

particles about the axis also vanishes1 
and as the sphere is made up of such    
rings1 the whole angular momentum is    
zero4 ,thus the condition is sufficient4
  ,thomson then proves that if the      
angular momentum thus considered be zero
 for every portion of the liquid at any 
one instant1 it remains zero at every   
subsequent instant2 that is1 no physical
 action whatsoever could set up angular 
momentum within the fluid1 which1 it is 
to be remembered1 is supposed to be     
frictionless4 ,the proof here given     
cannot be sketched because it depends on
 the differential equation of continuity
 satisfied by the velocity-potential    
throughout the fluid 7the same          
differential equation1 in fact1 that is 
satisfied by the distribution of        
temperature in a uniform conducting     
medium in the stationary state71 and the
 consequent expression of this function 
for any spherical space in the fluid as 
a series of spherical harmonic          
functions4 ,to a reader to whom the     

properties of these functions are known 
the process can present no difficulty4  
,an entirely different proof of this    
proposition is given subsequently in the
 paper1 and depends on a new and very   
general theorem1 which has been         
described as containing almost the whole
 theory of the motion of a fluid4 ,this 
depends on what ,thomson called the flow
 along any path joining any two points  
,p1 ,q in the fluid4 ,let q be the      
velocity of the fluid at any element of 
length ds of such a path1 and  be the   
angle between the direction of ds 7taken
 positive in the sense from ,p to ,q7   
and the direction of q3 q cos 4ds is the
 flow along ds4 ,if u1 v1 w be the      
components of q at ds1 parallel to the  
axes1 and dx1 dy1 dz be the projections 
of ds on the axes1 udx  vdy  wdz is the 
same thing as q cos 4ds4 ,the sum of the
 values of either of these expressions  
for all the elements of the path between
 ,p and ,q is the flow along the path4  
,the statement that u1 v1 w are the     

space-rates of variation of a function  
7of x1 y1 z7 parallel to the axes1 or   
that q cos  is the space-rate of        
variation of  along ds1 merely means    
that this sum is the same for whatever  
path may be drawn from ,p to ,q4 ,this1 
however1 is only the case when the paths
 are so taken that in each case the     
value of  returns after variation along 
a closed path to the value which it had 
at the starting point1 that is1 the     
closed path must be capable of being    
contracted to a point without passing   
out of space occupied by irrotationally 
moving fluid4  ,since the flow from ,p  
to ,q is the same for any two paths     
which fulfil this condition1 the flow   
from ,p to ,q by any one path and from  
,q to ,p by any other must be zero4 ,the
 flow round such a closed path is not   
zero if the condition is not fulfilled1 
and its value was called by ,thomson the
 circulation round the path4  ,the      
general theorem which he established may
 now be stated4 ,consider any path      

joining ,,pq1 and moving with the fluid1
 so that the line contains always the   
same fluid particles4 ,let u1 v1  be the
 time-rates of change of u1 v1 w at an  
element ds of the path1 at any instant1 
and du1 dv1 dw the excesses of the      
values of u1 v1 w at the terminal       
extremity of ds above the values at the 
other extremity2 then the time-rate of  
variation of udx  vdy  wdz is udx  vdy  
dz  udu  vdv  wdw or udx  vdy  dz  qdq1 
where q has the meaning specified above4
 ,thus if ,s be the flow for the whole  
path ,,pq1 and  its time-rate of        
variation1 ,s' denote the sum of udx    
vdy  dz along the path from ,p to ,q1   
and q1 q the resultant fluid velocities 
at ,q and ,p1 we get   ,s'  7q  q74     
,this is ,thomson's theorem4 ,if the    
curve be closed1 that is1 if ,p and ,q  
be coincident1 q  q and   ,s'4 ,but in  
certain circumstances ,s' is zero1 and  
so therefore is also 4 ,thus in the     
circumstances referred to1 as the closed
 path moves with the fluid  is          

continually zero1 and it follows that if
  is zero at any instant it remains zero
 ever after4 ,but  is only zero if u1 v1
 w are derivable from a potential1      
single valued in the space in which the 
closed path is drawn1 so that the path  
could be shrunk down to a point without 
ever passing out of such space4 ,in a   
perfect fluid if this condition is once 
fulfilled for a closed curve moving with
 the fluid1 it is fulfilled for this    
curve ever after4  ,the circumstances in
 which ,s' is zero are these3--the      
external force1 per unit mass1 acting on
 the fluid at any point is to be        
derivable from a potential-function1 and
 the density of the fluid is to be a    
function of the pressure 7also a        
function of the coordinates72 and these 
functions must be such as to render ,s' 
always zero for the closed path4 ,this  
condition is manifestly fulfilled in    
many important cases2 for example1 the  
forces are derivable from a potential   
due to actions1 such as gravity1 the    

origin of which is external to the      
fluid2 and the density is a function of 
the pressure 7in the present case it is 
a constant71 such that the part of ,s'  
which depends on pressure and density   
vanishes for the circuit4  ,it is to be 
clearly understood that the motion of a 
fluid may be irrotational although the  
value of ,s does not vanish for every   
closed path that can be drawn in it4    
,the fluid may occupy multiply          
continuous space1 and the path may or   
may not be drawn so that ,s shall be    
zero2 but what is necessary for         
irrotational motion within any space is 
that ,s should vanish for all paths     
which are capable of being shrunk down  
to zero without passing out of that     
space4 ,s need not vanish for a path    
which cannot be so shrunk down1 but it  
must1 if the condition just stated is   
fulfilled1 have the same value for any  
two paths1 one of which can be made to  
pass into the other by change of        
position without ever passing in whole  

or in part out of the space4 ,the       
potential is always single valued in    
fluid filling a singly continuous space 
such as that within a spherical shell1  
or between two concentric shells2 within
 a hollow anchor-ring the potential1    
though it exist1 and the motion be      
irrotational1 is not single valued4 ,in 
the latter case the motion is said to be
 cyclic1 in the former acyclic4  ,a     
number of consequences are deduced from 
this theorem2 and from these the        
properties of vortices1 which had       
previously been discovered by von       
,helmholtz1 immediately follow4 ,first  
take any surface whatever which has for 
bounding edge a closed curve drawn in   
the fluid1 and draw from any element of 
this surface1 of area d,s1 a line       
perpendicular to the surface towards the
 side chosen as the positive side1 and  
calculate the angular velocity 1 say1 of
 the fluid about that normal from the   
components of angular velocity          
determined in the manner explained at p4

 #afd4 ,this ,thomson called the        
rotation of the element4 ,now take the  
product d,s for the surface element4 ,it
 is easy to see that this is equal to   
half the circulation round the bounding 
edge of the element4 ,as the fluid      
composing the element moves the area d,s
 may change1 but the circulation round  
its edge by ,thomson's theorem remains  
unaltered4 ,thus  alters in the inverse 
ratio of d,s1 and the line drawn at     
right angles to the surface at d,s1 if  
kept of length proportional to 1 will   
lengthen or shorten as d,s contracts or 
expands4  ,now sum the values of d,s for
 the finite surface enclosed by the     
bounding curve4 ,it follows from the    
fact that d,s is equal to half the      
circulation round the edge of d,s1 that 
this sum1 which is usually denoted by   
d,s1 is equal to half the circulation   
round the closed curve which forms the  
edge of the surface4 ,also as the fluid 
moves the circulation round the edge    
remains unaltered1 and therefore so does

 also d,s for the elements enclosed by  
it4 ,it is important to notice that this
 sum being determined by the circulation
 in the bounding curve is the same for  
all surfaces which have the same        
boundary4  ,the equality of #b;d,s for  
the surface to the circulation round its
 edge was expressed by ,thomson as an   
analytical theorem of integration1 which
 was first given by ,stokes in a        
,smith's ,prize paper set in #ahed4 ,it 
is here stated1 apparently by an        
oversight1 that it was first given in   
,thomson and ,tait's ,natural           
,philosophy1  #aij4 ,in the second      
edition of the ,natural ,philosophy the 
theorem is attributed to ,stokes4 ,it is
 now well known as ,stokes's theorem    
connecting a certain surface integral   
with a line integral1 and has many      
applications both in physics and in     
geometry4  ,now consider the resultant  
angular velocity at any point of the    
fluid1 and draw a short line through    
that point in the direction of the axis 

of rotation4 ,that line may be continued
 from point to point1 and will coincide 
at every one of its points with the     
direction of the axis of rotation there4
 ,such an axial curve1 as it may be     
called1 it is clear moves with the      
fluid4 ,for take any infinitesimal area 
containing an element of the line2 the  
circulation round the edge of this area 
is zero1 since there is no rotation     
about a line perpendicular to the area4 
,hence the circulation along the axial  
curve is zero1 and the axial curves move
 with the fluid4  ,take now any small   
plane area d,s moving with the fluid1   
and draw axial lines through every point
 of its boundary4 ,these will form an   
axial tube enclosing d,s4 ,if  be the   
angle between the direction of resultant
 rotation and a perpendicular to d,s1   
the cross-section of the tube at right  
angles to the normal1 and to the axial  
lines which bound it1 is d,s4cos4 ,let  
these axial lines be continued in both  
directions from the element d,s4 ,they  

will enclose a tube of varying normal   
cross-section2 but the product of       
rotation and area of normal             
cross-section has everywhere the same   
value4 ,a vortex-tube with the fluid    
within it is called a vortex-filament4  
,it will be seen that this vortex-tube  
must be endless1 that is1 it must either
 return into itself1 or be infinitely   
long in one or both directions4 ,for if 
it were terminated anywhere within the  
fluid1 it would be possible to form a   
surface1 starting from a closed circuit 
round the tube1 continued along the     
surface of the tube to the termination1 
and then closed by a cap situated beyond
 the termination4 ,at no part of this   
surface would there be any rotation1 and
 d,s1 which is equal to the circulation1
 would be zero for it2 and of course    
this cannot be the case4 ,thus the tube 
cannot terminate within the fluid4 ,it  
can1 however1 have both of its ends on  
the surface1 or one on the bounding     
surface and the other at infinity1 if   

the fluid is infinitely extended in one 
direction1 but in that case the         
termination is only apparent4 ,the      
section is widened out at the surface2  
some of the bounding lines pass across  
to the other apparent termination1 when 
it also lies on the surface1 while the  
other lines pass off to infinity along  
the surface1 and correspond to other    
lines coming in from infinity to the    
other termination4 ,whether the surface 
is infinite or not1 the vortex is spread
 out into what is called a vortex-sheet1
 that is1 in a surface on the two sides 
of which the fluid moves with different 
tangential velocities4  ,through a      
vortex-ring or tube1 the fluid          
circulates in closed lines of flow1 each
 one of which is laced through the tube4
 ,the circulation along every line of   
flow which encloses the same system of  
vortex-tubes has the same value4  ,if   
any surface be drawn cutting a          
vortex-tube1 it is clear from the       
definition of the tube that the value of

 d,s for every such surface must be the 
same4 ,this ,thomson calls the rotation 
of the tube4  ,as was pointed out first 
by von ,helmholtz1 vortex-filaments     
correspond to circuits carrying currents
 and the velocity in the surrounding    
fluid to magnetic field-intensity4 ,the 
rotation of the tube corresponds to the 
strength of the current1 and sources and
 sinks to positive and negative magnetic
 poles4 ,thomson made great use of this 
analogy in his papers on                
electromagnetism4  ,examples of         
vortex-tubes are indicated on p4 #aed2  
and the reader may experiment with      
vortices in liquids with water in a     
tea-cup1 or in a river or pond1 at      
pleasure4 ,air vortices may be          
experimentally studied by means of a    
simple apparatus devised by ,professor  
,tait1 which may be constructed by      
anyone4  ,in one end of a packing-box1  
about #b;ft4 long by #ah;in4 wide and   
#ah;in4 deep1 a circular hole is cut1   
and the edges of the hole are thinned   

down to a blunt edge4 ,this can be      
closed at pleasure by a piece of board4 
,the opposite end is removed1 and a     
sheet of canvas stretched tightly in its
 place1 and tacked to the ends of the   
sides4 ,through two holes bored in one  
of the sides the mouths of two flasks   
with bent necks protrude into the box4  
,one of these flasks contains ammonia1  
the other hydrochloric acid4 ,when the  
hole at one end is closed up by a slip  
of tinplate1 and the liquids are heated 
with a spirit-lamp1 the vapours form a  
cloud of sal-ammoniac within the box1   
which is retained during its formation4 
,the hole is then opened1 and the canvas
 struck smartly with the palm of the    
open hand4 ,immediately a beautiful ring
 of smoke emerges1 clear-cut and        
definite as a solid1 and moves across   
the room4 7,see ,fig4 #ac47 ,of course1 
it is a ring of air1 made visible by the
 smoke carried with it4 ,by varying the 
shape of the aperture--for example1 by  
using instead of the hole cut in the    

wood1 a slide of tinplate with an       
elliptic hole cut in it--the            
vortex-rings can be set in vibration as 
they are created1 and the vibrations    
studied as the vortex moves4            
,illustration3 ,,fig4 #ac4  ,still more 
beautiful vortices can be formed in     
water by using a long tank of clear     
water to replace the air in which the   
vortex moves1 and a compartment at one  
end filled with water coloured with     
aniline1 instead of the smoke-box4 ,a   
hole in the dividing partition enables  
the vortex to be formed1 and a piston   
arrangement fitted to the opposite side 
enables the impulse to the water to be  
given from without4  ,from the account  
of the nature of vortex-motion given    
above1 it will be clear that vortices in
 a perfect fluid once existent must be  
ever existent4 ,to create a vortex      
within a mass of irrotationally moving  
perfect fluid is physically impossible4 
,it occurred to ,thomson1 therefore1    
that ordinary matter might be portions  

of a perfect fluid1 filling all space1  
differentiated from the surrounding     
fluid by the rotation which they        
possess4 ,such matter would fulfil the  
law of conservation1 as it could neither
 be created nor destroyed by any        
physical act4  ,the results of such     
experiments led ,thomson to frame his   
famous vortex-atom theory of matter1 a  
theory1 however1 which he felt          
ultimately was beset with so many       
difficulties as to be unworkable4  ,the 
paper on vortex-motion also deals with  
the modification of ,green's celebrated 
theorem of analysis1 which1 it was      
pointed out by ,helmholtz1 was necessary
 to adapt it to a space which is        
multiply continuous4 ,the theorem       
connects a certain volume-integral taken
 throughout a closed space with an      
integral taken over the bounding surface
 of the space4 ,this arises from the    
fact noticed above that in multiply     
continuous space 7for example1 the space
 within an endless tube7 the functions  

which are the subject of integration may
 not be single valued4 ,such a function 
would be the velocity-potential for     
fluid circulating round the tube--cyclic
 motion1 as it was called by ,thomson4  
,if a closed path of any form be drawn  
in such a tube1 starting from a point   
,p1 and doubling back so as to return to
 ,p without making the circuit of the   
tube1 the velocity-potential will vary  
along the tube1 but will finally return 
to its original value when the starting 
point is reached4 ,and the circulation  
round this circuit will be zero4 ,but if
 the closed path make the circuit of the
 tube1 the velocity-potential will      
continuously vary along the path1 until 
finally1 when ,p is reached again1 the  
value of the function is greater 7or    
less7 than the value assumed for the    
starting point1 by a certain definite   
amount which is the same for every      
circuit of the space4 ,if the path be   
carried twice round in the same         
direction1 the change of the function   

will be twice this amount1 and so on4   
,the space within a single endless tube 
such as an anchor-ring is doubly        
continuous2 but much more complicated   
cases can be imagined4 ,for example1 an 
anchor-ring with a cross-connecting tube
 from one side to the other would be    
triply continuous4  ,thomson showed that
 the proper modification of the theorem 
is obtained by imagining diaphragms     
placed across the space1 which are not  
to be crossed by any closed path drawn  
within the space1 and the two surfaces  
of each of which are to be reckoned as  
part of the bounding surface of the     
space4 ,one such diaphragm is sufficient
 to convert a hollow anchor-ring into a 
singly continuous space1 two would be   
required for the hollow anchor-ring with
 cross-connection1 and so on4 ,the      
number of diaphragms required is always 
one less than the degree of multiplicity
 of the continuity4  ,the paper also    
deals with the motion of solids in the  
fluid and the analogous motions of      

vortex-rings and their attraction by    
ordinary matter4 ,these can be studied  
with vortex-rings in air produced by the
 apparatus described above4 ,such a ring
 made to pass the re-entrant corner of a
 wall--the edge of a window recess1 for 
example--will appear to be attracted4 ,a
 large sphere such as a large           
terrestrial globe serves also very well 
as an attracting body4  ,two            
vortex-rings projected one after the    
other also act on one another in a very 
curious manner4 ,their planes are       
perpendicular to the direction of       
motion1 and the fluid is moving round   
the circular core of the ring4 ,there is
 irrotational cyclic motion of the fluid
 through the ring in one direction and  
back outside1 as shown in ,fig4 #ac1    
which can be detected by placing a      
candle flame in the path of the centre4 
,the first ring1 in consequence of the  
existence of that which follows it1     
moves more slowly1 and opens out more   
widely1 the following ring hastens its  

motion and diminishes in diameter1 until
 finally it overtakes the former and    
penetrates it4 ,as soon as it has passed
 through it moves ahead more and more   
slowly1 until the one which has been    
left behind begins to catch it up1 and  
the changes which took place before are 
repeated4 ,the one penetrating becomes  
in its turn the penetrated1 and so on in
 alternation4 ,great care and skill are1
 however1 necessary to make this        
interesting experiment succeed4  ,we    
have not space to deal here with other  
hydrodynamical investigations1 such as  
the contributions which ,thomson made to
 the discussion of the many difficult   
problems of the motion of solids through
 a liquid1 or to his very numerous and  
important contributions to the theory of
 waves4 ,the number and importance of   
his hydrodynamical papers may be judged 
from the fact that there are no less    
than fifty-two references to his papers1
 and thirty-five to ,thomson and ,tait's
 ,natural ,philosophy in the latest     

edition of ,lamb's ,hydrodynamics1 and  
that many of these are concerned with   
general theorems and results of great   
value4     ,,chapter ,x  ,,the ,,energy 
,,theory ,,of                           
,,electrolysis--,,electrical            
,,units--,,electrical ,,oscillations    
,,electrolysis ,,and ,,electrical       
,,units  ,in ,december #ahea ,thomson   
communicated an important paper to the  
,philosophical ,magazine on ,the        
,mechanical ,theory of ,electrolysis1   
and ,applications of ,mechanical ,effect
 to the ,measurement of ,electromotive  
,forces1 and of ,galvanic ,resistances1 
in ,absolute ,units4  ,in the first of  
these he supposed a machine of the kind 
imagined by ,faraday1 consisting of a   
metal disk1 rotating uniformly with its 
plane at right angles to the lines of   
force of a uniform magnetic field1 and  
touched at its centre and its           
circumference by fixed wires1 to send a 
current through an electrochemical      
apparatus1 to which the wires are       

connected4 ,a certain amount of work ,w 
was supposed to be spent in a given     
time1 during which a quantity of heat ,h
 was evolved in the circuit1 and a      
certain amount of work ,m spent in the  
chemical apparatus in effecting chemical
 change4 ,if ,h be taken in dynamical   
units1 ,w  ,h  ,m4  ,the work done in   
driving the disk1 if the intensity of   
the field is ,i1 the current produced c1
 the radius of the disc r1 and the      
angular velocity of turning w1 is ,ircw4
  ,thomson assumed that the work done in
 the electrochemical apparatus was equal
 to the heat of chemical combination of 
the substance or substances which       
underwent the chemical action1 taken    
with the proper sign according to the   
change1 if more compound substances than
 one were acted on4 ,hence ,m           
represented this resultant heat of      
combination4  ,the electrochemical      
apparatus was a voltameter containing a 
definite compound to be electrolysed1 or
 a voltaic cell or battery4 ,and by     

,faraday's experiments on electrolysis  
it was known that the amount of chemical
 action was proportional to the whole   
quantity of electricity passed through  
the cell in a given time1 so that the   
rate at which energy was being spent in 
the cell was at any instant proportional
 to the current at that instant4  ,the  
chemical change could be measured by    
considering only one of the elements set
 free1 or made to combine1 by the       
passage of the current1 and considering 
the quantity of heat 1 say1 for the     
whole chemical change in the cell       
corresponding to the action on unit mass
 of that element4 ,thus if ,e denote the
 whole quantity of that element operated
 on the heat of combination in the      
vessel was ,e4 ,if ,e be taken for unit 
of time1 and  denote the quantity set   
free by the passage of unit quantity of 
electricity1 then ,e  c1 since a current
 conveys c units of electricity in one  
second4 ,the number  is a definite      
quantity of the element1 and is called  

its electrochemical equivalent4 ,again1 
from ,joule's experiments1 ,h  ,rc1 if  
,r denote the resistance of the current1
 and so    ,ircw  ,rc  c  and    c      
7,irw  7,r  ,the quantity ,irw is the   
electromotive force due to the disk4    
,thus c was positive or negative        
according as ,irw was greater or less   
than 1 and was zero when ,irw  4 ,thus  
the electromotive force of the disk was 
opposed by a back electromotive force   
due to the chemical action in the       
voltameter or battery1 to which the     
wires from the disk were connected4     
,the conclusion arrived at therefore was
 that the electromotive force 7or1 as it
 was then termed1 the intensity7 of the 
electrochemical action was equal to the 
dynamical value of the whole chemical   
change effected by a current of unit    
strength in unit of time4  ,from this   
result ,thomson proceeded to calculate  
the electromotive forces required to    
effect chemical changes of different    
kinds1 and those of various types of    

voltaic cell4 ,supposing a unit of      
electricity to be carried by the current
 through the cell1 he considered the    
chemical changes which accompanied its  
passage1 and from the known values of   
heats of combination calculated their   
energy values4 ,in some parts the change
 was one of chemical combination1 in    
others one of decomposition of the      
materials1 and regard had to be paid to 
the sign of the heat-equivalent4 ,by    
properly summing up the whole           
heat-equivalents a net total was        
obtained which1 according to ,thomson1  
was the energy consumed in the passage  
of unit current1 and was therefore the  
electromotive force4 ,the theory was    
incomplete1 and required to be          
supplemented by thermodynamic theory1   
which shows that besides the            
electromotive force there must be       
included in the quantity set against the
 sum of heats a term represented by the 
product of the absolute temperature     
multiplied by the rate of variation of  

electromotive force with alteration of  
temperature4 ,thus the theory is only   
applicable when the electromotive force 
is not affected by variation of         
temperature4 ,the necessary addition    
here indicated was made by ,helmholtz4  
,in the next paper1 which appeared in   
the same number 7,december #ahea7 of the
 ,philosophical ,magazine1 the principle
 of work is applied to the measurement  
of electromotive forces and resistances 
in absolute units4 ,the advantages of   
such units are obvious4 ,nearly the     
whole of the quantitative work of the   
older experimenters was useless except  
for those who had actually made the     
observations3 it was hardly possible for
 one man to advance his researches by   
employing data obtained by others4 ,for 
the results were expressed by reference 
to apparatus and materials in the       
possession of the observers1 and to     
these others could obtain access only   
with great difficulty and at great      
expense--to say nothing of the          

uncertainty of comparisons made to      
enable the results of one man to be     
linked on to those made elsewhere1 and  
with other apparatus1 by another4 ,it   
was imperative1 therefore1 to obtain    
absolute units--units independent of    
accidents of place and apparatus--for   
the expression of currents1             
electromotive forces1 and resistances1  
so as to enable the results of the work 
of experiments all over the world to be 
made available to every one who read the
 published record4 7,see ,chap4 ,,xiii47
  ,the magneto-electric machine imagined
 in the former paper gave a means of    
estimating the electromotive force of a 
cell or battery in absolute units4 ,the 
same kind of machine is used here1 in   
the simpler form of a sliding conductor 
connecting a pair of insulated rails    
laid with their plane perpendicular to  
the lines of force of a uniform magnetic
 field4 ,if the rails be connected by a 
wire1 and the slider be moved so as to  
cut across the lines of force1 a current

 will be produced in the circuit4 ,the  
current can be measured in terms of the 
already known unit of current1 that     
current which flowing in a circle of    
radius unity produces a magnetic field  
at the centre of #b units4 ,this        
current1 c1 say1 in strength1 flowing in
 the circuit1 renders a dynamical force 
c,il necessary to move the slider of    
length l across the lines of force of   
the field of intensity ,i1 and if the   
speed of the slider required for the    
current c be v1 the rate at which work  
is done in moving the slider is c,ilv4  
,this must be the rate at which work is 
done in the circuit by the current1 and 
if the only work done be in the heating 
of the conductor1 we have c,ilv  ,rc1 or
 ,ilv  ,rc1 so that ,ilv is the         
electromotive force4 ,any electromotive 
force otherwise produced1 which gave    
rise to the same current1 must obviously
 be equal to ,ilv1 so that the unit of  
electromotive force can thus be properly
 defined4  ,thomson used a              

foot-grain-second system of units2 but  
from this arrangement are now obtained  
the ,c4,g4,s4 units of electromotive    
force and resistance4 ,if ,i is one     
,c4,g4,s4 unit1 l one centimetre1 and v 
one centimetre per second1 we have unit 
electromotive force in the ,c4,g4,s4    
system4 ,also in one ,c4,g4,s4 unit of  
resistance if c be unity as well as     
,ilv4  ,the idea of the determination of
 a resistance in absolute units on      
correct principles was due to ,w4       
,weber1 who also gave methods of        
carrying out the measurement2 and the   
first determination was made by         
,kirchhoff in #ahdi4 ,thomson appears1  
however1 to have been the first to      
discuss the subject of units from the   
point of view of energy4 ,this mode of  
regarding the matter is important1 as   
the absolute units are so chosen as to  
enable work done by electric and        
magnetic forces to be reckoned in the   
ordinary dynamical units4 ,a vast amount
 of experimental resource and skill has 

been spent since that time on the       
determination of resistance1 though not 
more than the importance of the subject 
warranted4 ,we shall have to return to  
the subject in dealing with the work of 
the ,british ,association on ,electrical
 ,standards1 of which ,thomson was for  
long an active member4   ,,electrical   
,,oscillations  ,in his famous tract on 
the conservation of energy1 published in
 #ahdg1 von ,helmholtz discussed some   
puzzling results obtained by ,riess in  
the magnetisation of iron wires by the  
current of a ,leyden jar discharge      
flowing in a coil surrounding them1 and 
by the fact1 observed by ,wollaston1    
that when water was decomposed by       
,leyden jar discharges a mixture of     
oxygen and hydrogen appeared at each    
electrode1 and suggested that possibly  
the discharge was oscillatory in        
character4  ,in #ahec the subject was   
discussed mathematically by ,thomson1 in
 a paper which was to prove fruitful in 
our own time in a manner then little    

anticipated4 ,the jar is given1 let us  
say1 with the interior coating charged  
positively1 and the exterior coating    
charged negatively4 ,a coil or helix of 
wire has its ends connected to the two  
coatings1 and a current immediately     
begins in the wire1 and gradually 7not  
slowly7 increases in strength4          
,accompanying the creation of the       
current is the production of a magnetic 
field1 that is1 the surrounding space is
 made the seat of magnetic action4 ,the 
magnetic field1 as we shall see from    
another investigation of ,thomson's1    
almost certainly involves motion in or  
of a medium--the ether--filling the     
space where the magnetic action is found
 to exist4 ,the charge of the jar       
consists of a state of intense and      
peculiar strain in the glass plate      
between the coatings4 ,when the plates  
are connected by the coil1 this state of
 strain breaks down and motion in the   
medium ensues1 not merely between the   
plates1 but also in the surrounding     

space--in fact1 in the whole field4     
,this motion--which is not to be        
confused with bodily displacement of    
finite parts of the medium--is opposed  
by something akin to inertia of the     
medium 7the property that confers energy
 on matter when in motion71 so that when
 the motion is started it persists1     
until it is finally wiped out by        
resistance of the nature of friction4   
,the inertia here referred to depends on
 the mode in which the coil is wound1 or
 whether it contains or not an iron     
core4  ,if the work done in charging a  
,leyden jar or electric condenser1 by   
bringing the charge to the condenser in 
successive small portions1 is           
considered1 it is at once clear that it 
must be proportional to the square of   
the whole quantity of electricity       
brought up4 ,for whatever the charge may
 be1 let it be brought up from a great  
distance in a large number ,n of equal  
instalments4 ,the larger the whole      
amount the larger must each instalment  

be1 and therefore the greater the amount
 accumulated on the condenser when any  
given number of instalments have been   
deposited4 ,but the greater any charge  
that is being brought up1 and also the  
greater the charge that has already     
arrived1 the greater is the repulsion   
that must be overcome in bringing up    
that instalment1 in simple proportion in
 each case1 and therefore the greater   
the work done4 ,thus the whole work done
 in bringing up the charge must be      
proportional to ,q4 ,we suppose it to be
 ,q,c1 where ,c is a constant depending 
on the condenser and called its         
capacity4  ,the idea of the charge as a 
quantity of some kind of matter1 brought
 up and placed on the insulated plate of
 the condenser1 has only a              
correspondence to the fact1 which is    
that the medium between the plates is   
the seat1 when the condenser is charged1
 of a store of energy1 which can only be
 made available by connecting the plates
 of the condenser by a wire or other    

conductor4 ,the charge is only a surface
 aspect of the state of the medium1     
apparently a state of strain1 to which  
the energy belongs4  ,when a wire is    
used to connect the plates the state of 
strain disappears2 the energy comes out 
from the medium between the plates by   
motion sideways of the tubes of strain  
7so that the insulating medium is under 
longitudinal tension and lateral        
pressure7 which1 according to ,faraday's
 conception of lines of electric force  
connecting the charge on a body with the
 opposite charges on other bodies1 run  
from plate to plate1 when the condenser 
is in equilibrium in the changed state4 
,these tubes move out with their ends on
 the wire1 carrying the energy with     
them1 and the ends run towards one      
another along the wire2 the tube        
shortens in the process1 and energy is  
lost in the wire4 ,the ends of a tube   
thus moving represent portions of the   
charges which were on the plates1 and   
the oppositely-directed motions of the  

opposite charges represent a current    
along the wire from one conductor to the
 other4 ,the motion of the tubes is     
accompanied by the development of a     
magnetic field1 the lines of force of   
which are endless1 and the direction of 
which at every point is perpendicular at
 once to the length of the tube and to  
the direction in which it is there      
moving4 ,in certain circumstances the   
tube1 by the time its ends have met1    
will have wholly disappeared in the     
wire1 and the whole energy will have    
gone to heat the wire3 in other         
circumstances the ends will meet before 
the tube has disappeared1 the ends will 
cross1 and the tube will be carried back
 to the condenser and reinserted in the 
opposite direction4 ,at a certain time  
this will have happened to all the      
tubes1 though they will have lost some  
of their energy in the process2 and the 
condenser will again be charged1 though 
in the opposite way to that in which it 
was at first4 ,then the tubes will move 

out again1 and the same process will be 
repeated3 once more the condenser will  
be charged1 but in the same direction as
 at first1 and once more with a certain 
loss of energy4 ,again the process of   
discharge and charge will take place1   
and so on1 again and again1 until the   
whole energy has disappeared4 ,this     
process represents1 according to the    
modern theory of the flow of energy in  
the electromagnetic field1 with more or 
less accuracy1 what takes place in the  
oscillatory discharge of a condenser4   
,the motion of the tubes with their ends
 on the wire represents a certain amount
 of energy1 commonly regarded as        
kinetic1 and styled electrokinetic      
energy4 ,if c denote the current1 that  
is1 the rate1 -d,qdt1 at which the      
charge of the condenser is being        
changed1 and ,l a quantity called       
self-inductance1 depending mainly on the
 arrangement of the connecting          
wire--whether it is wound in a coil or  
helix1 with or without an iron core1 or 

not--the electrokinetic energy will be  
,lc4 ,this is analogous to the kinetic  
energy mv of a body 7say a pendulum bob7
 of mass m and velocity v1 so that ,l   
represents a quantity for the conducting
 arrangement analogous to inertia1 and c
 is the analogue of the velocity of the 
body4 ,the whole energy at any instant  
is thus    ,q,c  ,lc1 or ,q,c           
,l7d,qdt74  ,the loss of energy due to  
heating of the conducting connection is 
not completely understood1 though its   
quantitative laws have been quite fully 
ascertained and expressed in terms of   
magnitudes that are capable of          
measurement4 ,it was found by ,joule to 
be proportional to the second power1 or 
square1 of the current1 and to a        
quantity ,r depending on the conductor1 
and called its resistance4 ,the         
generation of heat in the conductor     
seems to be due to some kind of         
frictional action of particles of the   
conductor set up by the penetration of  
the ,faraday tubes into it4 ,a conductor

 is unable to bear any tangential action
 exerted upon it by ,faraday tubes1     
which1 however1 when they exist1 begin  
and end at material particles1 except   
when they are endless1 as they may be in
 the radiation of energy4 ,when the     
,faraday tubes are moving with any      
ordinary speed they are not at their    
ends perpendicular to the conducting    
surface from which they start or at     
which they terminate1 but are there more
 or less inclined to the surface1 and   
consequently there is tangential action 
which appears to displace the particles 
7not merely at the surface1 unless the  
alternation is very rapid7 relatively to
 one another and so cause frictional    
generation of heat4  ,the time rate of  
generation of heat is thus ,rc1 or      
,r7d,qdt71 when the units in which ,r   
and c are expressed are such as to make 
this quantity a rate of doing work in   
the true dynamical sense4 ,this is the  
rate at which the sum of energy already 
found is being diminished1 and so the   

equation    d_/dt7,q,c7  ,l7d,qdt7      
,r7d,qdt7  holds1 or leaving out the    
common factor d,qdt1 the equation       
,l7d,qdt7  ,r7d,qdt7  ,q,c  #j  ,this   
last equation was established by        
,thomson1 and is precisely that which   
would be obtained for a pendulum bob of 
mass ,l1 pulled back towards the        
position of equilibrium with a force    
,q,c1 where ,q is the displacement from 
the middle position1 and having its     
motion damped out by resisting force of 
amount ,r per unit of the velocity4  ,it
 is more instructive perhaps to take the
 oscillatory motion of a spiral spring  
hung vertically with a weight on its    
lower end1 as that which has a          
differential equation equivalent to the 
equation just found4 ,when the stretch  
is of a certain amount1 there is        
equilibrium--the action of the spring   
just balances the weight1--and if the   
spring be stretched further there will  
be a balance of pull developed tending  
to bring the system back towards the    

equilibrium position4 ,if left to itself
 the system gets into motion1 which1 if 
the resistance is not too great1 is     
added to until the equilibrium position 
is reached2 and the motion1 which is    
continued by the inertia of the mass1   
only begins to fall off as that position
 is passed1 and the pull of the spring  
becomes insufficient to balance the     
weight4 ,thus the mass oscillates about 
the position of equilibrium1 and the    
oscillations are successively smaller   
and smaller in extent1 and die out as   
their energy is expended finally in     
doing work against friction4  ,if the   
resisting force for finite motion is    
very great1 as for example when the     
vibrating mass of the pendulum or spring
 is immersed in a very viscous fluid1   
like treacle1 oscillation will not take 
place at all4 ,after displacement the   
mass will move at first fairly quickly1 
then more and more slowly back to the   
position of equilibrium1 which it will1 
strictly speaking1 only exactly reach   

after an infinite time4 ,the resisting  
force is here indefinitely small for an 
indefinitely small speed1 but it becomes
 so great when any motion ensues1 that  
as the restoring force falls off with   
the displacement1 no work is finally    
done by it1 except to move the body     
through the resisting medium4  ,the     
differential equation is applicable to  
the spring if ,q is again taken as      
displacement from the equilibrium       
position1 ,l as the inertia of the      
vibrating body1 #a;,c as the pull       
exerted by the spring per unit of its   
extension 7that is1 the stiffness of the
 spring71 and ,r has the same meaning as
 before4  ,in this case of motion1 as   
well as in that of the pendulum1 energy 
is carried off by the production of     
waves in the medium in which the        
vibrator is immersed4 ,these are        
propagated out from the vibrator as     
their source1 but no account of them is 
taken in the differential equation1     
which in that respect is imperfect4     

,there is no difficulty1 only the       
addition of a little complication1 in   
supplying the omission4  ,the formation 
of such waves by the spiral spring      
vibrator can be well shown by immersing 
the vibrating body in a trough of water1
 and the much greater rate of damping   
out of the motion in that case can then 
be compared with the rate of damping in 
air4  ,it has been indicated that the   
differential equation does not represent
 oscillatory motion if the value of ,r  
is too great4 ,the exact condition      
depends on the roots of the quadratic   
equation ,lx  ,rx  #a;,c  #j1 obtained  
by writing #a for ,q1 and x for ddt1 and
 then treating x as a quantity4 ,these  
roots are ,r#b,l  7,r#d,l  #a;,,cl71 and
 are therefore real or imaginary        
according as #d,l,c is less or greater  
than ,r4 ,if the roots are real1 that   
is1 if ,r be greater than #d,l,c1 the   
discharge will not be oscillatory2 the  
,faraday tubes referred to above will be
 absorbed in the wire without any return

 to the condenser4 ,the corresponding   
result happens with the vibrator when ,r
 is sufficiently great1 or ,l,c         
sufficiently small 7a weak spring and a 
small mass1 or both71 to enable the     
condition to be fulfilled4  ,if1        
however1 the roots of the quadratic are 
imaginary1 that is1 if #d,l,c be greater
 than ,r 7a condition which will be     
fulfilled in the spring analogue1 by    
making the spring sufficiently stiff and
 the mass large enough to prevent the   
friction from controlling the motion7   
the motion is one in which ,q disappears
 by oscillations about zero1 of         
continually diminishing amplitude4 ,a   
complete discussion gives for the period
 of oscillation #d,l7#d,l,c  ,r71 or if 
,r be comparatively small1 #b7,,lc74    
,the charge ,q falls off by the fraction
 e,,rt#b,l 7where e is the number       
#b4gahbh4447 in each period ,t1 and so  
gradually disappears4  ,thus electric   
oscillations are produced1 that is to   
say1 the charged state of the condenser 

subsides by oscillations1 in which the  
charged state undergoes successive      
reversals1 with dissipation of energy in
 the wire2 and both the period and the  
rate of dissipation can be calculated if
 ,l1 ,c1 and ,r are known1 or can be    
found1 for the system4 ,these quantities
 can be calculated and adjusted in      
certain definite cases1 and as the      
electric oscillations can be            
experimentally observed1 the theory can 
be verified4 ,this has been done by     
various experimenters4  ,returning to   
the pendulum illustration1 it will be   
seen that the pendulum held deflected is
 analogous to the charged jar1 letting  
the pendulum go corresponds to          
connecting the discharging coil to the  
coatings1 the motion of the pendulum is 
the analogue of that motion of the      
medium in which consists the magnetic   
field1 the friction of the air answers  
to the resistance of the wire which     
finally damps out the current4 ,the     
inertia or mass of the bob is the       

analogue of what ,thomson called the    
electromagnetic inertia of the coil and 
connections2 what is now generally      
called the self-inductance of the       
conducting system4 ,the component of    
gravity along the path towards the      
lowest point1 answers to the reciprocal1
 #a;,c1 of the capacity of the          
condenser4  ,it appears from the analogy
 that just as the oscillations of a     
pendulum can be prevented by immersing  
the bob in a more resisting medium1 such
 as treacle or oil1 so that when        
released the pendulum slips down to the 
vertical without passing it1 so by      
properly proportioning the resistance in
 the circuit to the electromagnetic     
inertia of the coil1 oscillatory        
discharge of the ,leyden jar may also be
 rendered impossible4  ,all this was    
worked out in an exceedingly instructive
 manner in ,thomson's paper2 the account
 of the matter by the motion of ,faraday
 tubes is more recent1 and is valuable  
as suggesting how the inertia effect of 

the coil arises4 ,the analogy of the    
pendulum is a true one1 and enables the 
facts to be described2 but it is to be  
remembered that it becomes evident only 
as a consequence of the mathematical    
treatment of the electrical problem4    
,the paper was of great importance for  
the investigation of the electric waves 
used in wireless telegraphy in our own  
time4 ,it enabled the period of         
oscillation of different systems to be  
calculated1 and so the rates of exciters
 and receivers of electric waves to be  
found4 ,for such vibrators are really   
,leyden jars1 or condensers1 caused to  
discharge in an oscillatory manner4     
,this application was not foreseen by   
,thomson1 and1 indeed1 could hardly be1 
as the idea of electric waves in an     
insulating medium came a good deal later
 in the work of ,maxwell4 ,yet the      
analogy of the pendulum1 if it had then 
been examined1 might have suggested such
 waves4 ,as the bob oscillates backwards
 and forwards the air in which it is    

immersed is periodically disturbed1 and 
waves radiate outwards from it through  
the surrounding atmosphere4 ,the energy 
of these waves is exceedingly small1    
otherwise1 as pointed out above1 a term 
would have to be included in the theory 
of the resisted motion of the pendulum  
to account for this energy of radiation4
 ,so likewise when the electric         
vibrations proceed1 and the insulating  
medium is the seat of a periodically    
varying magnetic field1 electromagnetic 
waves are propagated outwards through   
the surrounding medium--the ether--and  
the energy carried away by the waves is 
derived from the initial energy of the  
charged condenser4 ,in strictness also  
,thomson's theory of electric           
oscillations requires an addition to    
account for the energy lost by          
radiation4 ,this is wanting1 and the    
whole decay of the amount of energy     
present at the oscillator is put down to
 the action of resistance--that is1 to  
something of the nature of frictional   

retardation4 ,notwithstanding this      
defect of the theory1 which is after all
 not so serious as certain difficulties 
of exact calculation of the             
self-inductance of the discharging      
conductor1 the periods of vibrators can 
be very accurately found4 ,when these   
are known it is only necessary to       
measure the length of an electrical wave
 to find its velocity of propagation4   
,when electromagnetic waves were        
discovered experimentally in #ahhh by   
,heinrich ,hertz1 it was thus that he   
was able to demonstrate that they       
travelled with the velocity of light4   
,thomson suggested that double1 triple  
and quadruple flashes of lightning might
 be successive flashes of an oscillatory
 discharge4 ,he also pointed out that if
 a spark-gap were included in a properly
 arranged condenser and discharging     
wire1 it might be possible1 by means of 
,wheatstone's revolving mirror1 to see  
the sparks produced in the successive   
oscillations1 as points or short lines  

of light separated by dark intervals1   
instead of a single point of light1 or  
of an unbroken line of light1 as it     
would be if the discharge were          
instantaneous1 or were continuous1 or of
 appreciable duration4  ,this           
anticipation was verified by experiments
 made by ,feddersen1 and published in   
#ahei 7,pogg4 ,ann41 #ajh1 #ahei74 ,the 
subject was also investigated in        
,helmholtz's laboratory at ,berlin1 by  
,n4 ,schiller1 who1 determining the     
period for condensers with different    
substances between the plates1 was able 
to deduce the inductive capacities of   
these substances 7,pogg4 ,ann41 #aeb1   
#ahgd74 ,the specific inductive capacity
 of an insulator is the ratio of the    
capacity of a condenser with the        
substance between the plates to the     
capacity of an exactly similar condenser
 with air between the plates4  ,the     
particular case of non-oscillatory      
discharge obtained by supposing ,c and  
,q both infinitely great and to have a  

finite ratio ,v 7which will be the      
potential1 p4 #cd1 of the charged       
plate71 is considered in the paper4 ,the
 discharging conductor is thus subjected
 to a difference of potential suddenly  
applied and maintained at one end1 while
 the other end is kept at potential     
zero4 ,the solution of the differential 
equation for this case will show how the
 current rises from zero in the wire to 
its final steady value4 ,if c be put as 
before for the current d,qdt1 and the   
constant value ,v for ,q,c1 the equation
 is    ,l7dcdt7  ,rc  ,v  which gives1  
since c  #j when t  #j1    c  7,v,r7#a  
e7,r,l7t4  ,thus1 when an infinite time 
has elapsed the current has become ,v,r1
 the steady value4  ,thomson concludes  
by showing how1 by measuring the        
non-oscillatory discharge of a condenser
 7the capacity of which can be          
calculated7 by means of an              
electrodynamometer and an ordinary      
galvanometer arranged in series1 what   
,w4 ,weber called the duration of the   

discharging current may be determined4  
,from this ,thomson deduced a value for 
the ratio of the electromagnetic unit of
 electricity to the electrostatic unit1 
and indicated methods of determining    
this ratio experimentally4 ,this ratio  
is of fundamental importance in         
electromagnetic theory1 and is          
essentially of the nature of a speed4   
,according to ,maxwell it is the speed  
of propagation of electromagnetic waves 
in an insulating medium for which the   
units are defined4 ,it was first        
determined in the ,glasgow laboratory by
 ,mr4 ,dugald ,mc,kichan1 and has been  
determined many times since4 ,it is     
practically identical with the speed of 
light as ascertained by the best        
experiments4     ,,chapter ,,xi         
,,thomson ,,and ,,tait',s ',,natural    
,,philosophy'--,,gyrostatic             
,,action--',,electrostatics ,,and       
,,magnetism'   ,,the ',,natural         
,,philosophy'  ,professor ,tait was     
appointed to the ,chair of ,natural     

,philosophy in the ,university of       
,edinburgh in #ahfj1 and came almost    
immediately into frequent contact with  
,thomson4 ,both were ,peterhouse men1   
trained by the same private             
tutor--,william ,hopkins--both were     
enthusiastic investigators in           
mathematical as well as in experimental 
physics1 they taught in the sister      
universities of ,edinburgh and ,glasgow1
 and had much the same kind of classes  
to deal with and the same educational   
problems to solve4 ,tait was an         
,edinburgh man--an old school-fellow of 
,clerk ,maxwell at the ,edinburgh       
,academy--and had therefore been exposed
 to that contact1 in play and in work1  
with compeers of like age and           
capabilities1 which is one of the best  
preparations for the larger school and  
more serious struggles of life4         
,thomson's early education1 under his   
father's anxious care1 had no doubt     
certain advantages1 and his early       
entrance into college classes gave him  

to a great extent that intercourse with 
others for which such advantages are    
never complete compensation4 ,the two   
men had much community of thought and   
experience1 and the literary partnership
 into which they entered was hailed as  
one likely to do much for the progress  
of science4  ,in some ways1 however1    
,thomson and ,tait were very different  
personalities4 ,thomson troubled himself
 little with metaphysical subtleties1   
his conceptions were like those of      
,newton1 absolutely clear so far as they
 went2 he never1 in his teaching at     
least1 showed any disposition to discuss
 the foundations of dynamics1 or the    
conception of motion in a straight line4
 ,these were taken for granted like the 
fundamental ideas in a book on geometry2
 and the student was left to do what    
every true dynamical student must do for
 himself sooner or later--to compare the
 abstractions of dynamics with the      
products of his experience in the world 
of matter and force4 ,perhaps a little  

guidance now and then in the            
difficulties about conceptions1 which   
beset every beginner1 might not have    
been amiss3 but ,thomson was so intent  
on the concrete example in              
hand--pendulum or gyrostat1 or what     
not--that he left each man to form or   
correct his own ideas by the lessons    
which such examples afford to every one 
who carefully examines them4  ,tait1 on 
the other hand1 though he continually   
denounced metaphysical discussion1 was  
in reality much more metaphysical than  
,thomson1 and seemed to take pleasure in
 the somewhat transcendental arguments  
with regard to matters of analysis which
 were put forward1 especially in the    
,elements of ,quaternions1 by ,sir      
,william ,rowan ,hamilton1 of ,dublin1 a
 master whom he much revered4 ,but there
 is metaphysics and metaphysics6 and the
 pronouncements of professed            
metaphysicians were often characterised 
as non-scientific and fruitless1 which  
no doubt they were from the physical    

point of view4  ,then ,tait was strongly
 convinced of the importance for physics
 of the quaternion analysis3 ,thomson   
was not1 to say the least2 and this was 
probably the main reason why the        
vectorial treatment of displacement1    
velocities1 and other directed          
quantities1 has no place in the joint   
writings of the two ,scottish           
professors4 ,in controversy ,tait was a 
formidable antagonist3 when war was     
declared he gave no quarter and asked   
for none1 though he never fought an     
unchivalric battle4 ,he admired foreign 
investigators--and especially von       
,helmholtz--but he was always ready to  
put on his armour and place lance in    
rest for the cause of ,british science4 
,thomson was much less of a combatant1  
though he also could bravely splinter a 
spear with an opponent on occasion1 as  
in the memorable discussion with ,huxley
 on the ,age of the ,earth4  ,tait's    
professorial lectures were always models
 of clear and logical arrangement4      

,every statement bore on the business in
 hand2 the experimental illustrations1  
always carefully prepared beforehand1   
were called for at the proper time and  
were invariably successful4 ,with       
,thomson it was otherwise3 his          
digressions1 though sometimes inspired  
and inspiring1 were fatal to the success
 of the utmost efforts of his assistants
 to make his lectures successful        
systematic expositions of the facts and 
principles of elementary physics4  ,as  
has been stated in ,chapter ,,iv1 two   
books were announced in #ahfc as in     
course of preparation for the ensuing   
session of ,college4 ,these were not    
published until #ahfg and #ahgc2 the    
first issued was the famous ,treatise on
 ,natural ,philosophy1 the second was   
entitled ,elements of ,natural          
,philosophy1 and consisted in the main  
of part of the non-mathematical or large
 type portions of the ,treatise4 ,the   
scheme of the latter was that of an     
articulated skeleton of statements of   

principles and results1 printed in      
ordinary type1 with the mathematical    
deductions and proofs in smaller type4  
,as was to be expected1 the ,elements1  
to a student whose mathematical reading 
was wide enough to tackle the ,treatise1
 was the more difficult book of the two 
to completely master4 ,but the continued
 large print narrative1 as it may be    
called1 is extremely valuable4 ,it is a 
memorial of a habit of mind which was   
characteristic of both authors4 ,they   
kept before them always the idea or     
thing rather than its symbol2 and thus  
the edifice which they built up seemed  
never obscured by the scaffolding and   
machinery used in its erection4 ,and as 
far as possible in processes of         
deduction the ideas are emphasised      
throughout2 there is no mere putting in 
at one end and taking out at the other2 
the result is examined and described at 
every stage4 ,as in all else of         
,thomson's work1 physical interpretation
 is kept in view at every step1 and made

 available for correction and avoidance 
of errors1 and the suggestion of new    
inquiries4  ,the book as it stands      
consists of ,division ,i1 ,preliminary  
and part of ,division ,,ii1 ,abstract   
,dynamics4 ,division ,i includes the    
chapter on ,kinematics already referred 
to1 a chapter on ,dynamical ,laws and   
,principles1 chapters on ,experience and
 ,measures and ,instruments4 ,division  
,,ii is represented only by ,chapter ,v1
 ,introductory2 ,chapter ,,vi1 ,statics 
of a ,particle and ,attractions2 and    
,chapter ,,vii1 ,statics of ,solids and 
,fluids4 ,thus ,abstract ,dynamics is   
without the more complete treatment of  
,kinetics to which1 as well as to       
,statics1 the discussion of ,dynamical  
,laws and ,principles was intended to be
 an introduction4 ,but to a considerable
 extent1 as we shall see1 ,kinetics is  
treated in this introductory chapter3   
indeed1 the discussion of the general   
theorems of dynamics and their          
applications to kinetics is remarkably  

complete4  ,in ,volume ,,ii it was      
intended to include chapters on the     
kinetics of a particle and of solid and 
fluid bodies1 on the vibrations of solid
 bodies1 and on wave-motion in general4 
,it was expected also to contain a      
chapter much referred to in ,volume ,i1 
on ,properties of ,matter4 ,that the    
work was not completed is a matter of   
keen regret to all physicists1 regret1  
however1 now tempered by the fact that  
many of the subjects of the unfulfilled 
programme are represented by such works 
as ,lord ,rayleigh's ,theory of ,sound1 
,lamb's ,hydrodynamics1 and ,routh's    
,dynamics of a ,system of ,rigid        
,bodies4 ,but all deeply lament the loss
 of the ,properties of ,matter4 ,no one 
can ever write it as ,thomson would have
 written it4 ,his students obtained in  
his lectures glimpses of the things it  
might have contained1 and it was most   
eagerly looked for4 ,if that chapter    
only had been given1 the loss caused by 
the discontinuance of the book would not

 have been so irreparable4  ,the first  
edition of the book was published by the
 ,clarendon ,press1 ,oxford4 ,it was    
printed by ,messrs4 ,constable1 of      
,edinburgh1 and is a beautiful specimen 
of mathematical typography4 ,in some    
ways the first edition is exceedingly   
interesting1 for it is not too much to  
say that its issue had an influence on  
dynamical science1 and its exposition in
 this country1 only second to that due  
to ,newton's ,principia4 ,three other   
works1 perhaps1 have had the same degree
 and kind of influence on mathematical  
thought--,laplace's ,mcanique ,cleste1  
,lagrange's ,mcanique ,analytique1 and  
,fourier's ,thorie ,analytique de la    
,chaleur4  ,the second edition was      
issued by the ,cambridge ,university    
,press as ,parts ,i and ,,ii in #ahgh   
and #ahhc4 ,various younger             
mathematicians now of                   
eminence--,professor ,chrystal1 of      
,edinburgh1 and ,professor ,burnside1 of
 ,greenwich1 may be mentioned--read the 

proofs1 and it is on the whole          
remarkably free from typographical and  
other errors4 ,with the issue of ,part  
,,ii1 the continuation was definitely   
abandoned4  ,in the second edition many 
topics are more fully discussed1 and the
 contents include a very valuable       
account of cycloidal motion 7or         
oscillatory motion1 as it is more       
usually called71 and of a revised       
version of the chapter on ,statics which
 forms the concluding portion of the    
book1 and which discusses some of the   
great problems of terrestrial and       
cosmical physics4  ,various speculations
 have been indulged in1 from time to    
time1 as to the respective parts        
contributed to the work by the two      
authors1 but these are generally very   
wide of the mark4 ,the mode of          
composition of the sections on cycloidal
 7oscillatory7 motion gives some idea of
 ,thomson's method of working4 ,his     
proofs 7of ,t and ,t-dash as the authors
 called the book7 were carried with him 

by rail and steamer1 and he worked      
incessantly 7without1 however1          
altogether withdrawing his attention    
from what was going on around him67 at  
corrections and additions4 ,he corrected
 heavily on the proofs1 and then        
overflowed into additional manuscript4  
,thus1 when he came to the short        
original  #cdc1 he greatly extended that
 in the first instance1 and proceeded   
from section to section until additions 
numbered from  #cdc;a to  #cdcp1        
amounting in all to some ten pages of   
small print1 had been interpolated4     
,similarly  #cde was extended by the    
addition of  #cde 7i7 to #cde 7xxviii71 
mainly on gyrostatic domination4 ,the   
method had the disadvantage of          
interrupting the printers and keeping   
type long standing1 but the matter was  
often all the more inspiring through    
having been produced under pressure from
 the printing office4 ,indeed1 much was 
no doubt written in this way which1 to  
the great loss of dynamical science1    

would otherwise never have been written 
at all4  ,the kinematical discussion    
begins with the consideration of motion 
along a continuous line1 curved or      
straight4 ,this naturally suggests the  
ideas of curvature and tortuosity1 which
 are fully dealt with mathematically1   
before the notion of velocity is        
introduced4 ,when that is done1 the     
directional quality of velocity is not  
so much insisted on as is now the case3 
for example1 a point is spoken of as    
moving in a curve with a uniform        
velocity2 and of course in the language 
of the present time1 which has been     
rendered more precise by vector ideas1  
if not by vector-analysis1 the velocity 
of a point which is continually changing
 the direction of its motion1 cannot be 
uniform4 ,the same remark may be made   
regarding the treatment of acceleration3
 in both cases the reference of the     
quantity to three ,cartesian axes is    
immediate1 and the changes of the       
components1 thus fixed in direction1 are

 alone considered4  ,there can be no    
doubt that greater clearness is obtained
 by the process afterwards insisted on  
by ,tait1 of considering by a           
hodographic diagram the changes of      
velocity in successive intervals of     
time1 and from these discovering the    
direction and magnitude of the rate of  
change at each instant4 ,this method is 
indeed indicated at  #cg1 but no diagram
 is given1 and the properties of the    
hodograph are investigated by means of  
,cartesians4 ,the subject is1 however1  
treated in the ,elements by the method  
here indicated4  ,remarkable features of
 this chapter are the very complete     
discussion of simple harmonic or        
vibratory motion1 the sections on       
rotation1 and the geometry of rolling   
and precessional motion1 and on the     
curvature of surfaces as investigated by
 kinematical methods4 ,a remark made in 
#if should be borne in mind by all who  
essay to solve gyrostatic problems4 ,it 
is that just as acceleration1 which is  

always at right angles to the motion of 
a point1 produces a change in the       
direction of the motion but none in the 
speed of the point 7it does influence   
the velocity71 so an action1 tending    
always to produce rotation about an axis
 at right angles to that about which a  
rigid body is already rotating1 will    
change the direction of the axis about  
which the body revolves1 but will       
produce no change in the rate of        
turning4#bj  ,a very full and clear     
account of the analysis of strains is   
given in this chapter1 in preparation   
for the treatment of elasticity which   
comes later in the book2 and a long     
appendix is added on ,spherical         
,harmonics1 which are defined as        
homogeneous functions of the coordinates
 which satisfy the differential equation
 of the distribution of temperature in a
 medium in which there is steady flow of
 heat1 or of distribution of potential  
in an electrical field4 ,this appendix  
is within its scope one of the most     

masterly discussions of this subject    
ever written1 though1 from the point of 
view of rigidity of proof1 required by  
modern function-theory1 it may be open  
to objection4  ,in the next chapter1    
which is entitled ,dynamical ,laws and  
,principles1 the authors at the outset  
declare their intention of following the
 ,principia closely in the discussion of
 the general foundations of the subject4
 ,accordingly1 after some definitions   
the laws of motion are stated1 and the  
opportunity is taken to adopt and       
enforce the ,gaussian system of absolute
 units for dynamical quantities4 ,as has
 been indicated above1 the various      
difficulties more or less metaphysical  
which must occur to every thoughtful    
student in considering ,newton's laws of
 motion are not discussed1 and probably 
such a discussion was beyond the scheme 
which the authors had in view4 ,but     
metaphysics is not altogether excluded4 
,it is stated that matter has an innate 
power of resisting external influences1 

so that every body1 as far as it can1   
remains at rest1 or moves uniformly in a
 straight line1 and it is stated that   
this property--inertia--is proportional 
to the quantity of matter in the body4  
,this statement is criticised by        
,maxwell in his review of the ,natural  
,philosophy in ,nature in #ahgi 7one of 
the last papers that ,maxwell wrote74   
,he asks1 ,is it a fact that 'matter has
 any power1 either innate or acquired1  
of resisting external influences'8 ,does
 not every force which acts on a body   
always produce that change in the motion
 of the body by which its value1 as a   
force1 is reckoned8 ,is a cup of tea to 
be accused of resisting the sweetening  
influence of sugar1 because it          
persistently refuses to turn sweet      
unless the sugar is put into it8  ,this 
innate power of resisting is merely the 
materi vis insita of ,newton's          
,definitio ,,iii1 given in the          
,principia1 and the statement to which  
,maxwell objects is only a free         

translation of that definition4         
,moreover1 when a body is drawn or      
pushed by other bodies1 it reacts on    
those bodies with an equal force1 and   
this reaction is just as real as the    
action3 its existence is due to the     
inertia of the body4 ,the definition1   
from one point of view1 is only a       
statement of the fact that the          
acceleration produced in a body in      
certain circumstances depends upon the  
body itself1 as well as on the other    
bodies concerned1 but from another it   
may be regarded as accounting for the   
reaction4 ,the mass or inertia of the   
body is only such a number that1 for    
different bodies in the same            
circumstances as to the action of other 
bodies in giving them acceleration1 the 
product of the mass and the acceleration
 is the same for all4 ,it is1 however1 a
 very important property of the body1   
for it is one factor of the quantum of  
kinetic energy which the body           
contributes to the energy of the system1

 in consequence of its motion relatively
 to the chosen axes of reference1 which 
are taken as at rest4  ,the relativity  
of motion is not emphasised so greatly  
in the ,natural ,philosophy as in some  
more modern treatises1 but it is not    
overlooked2 and whatever may be the view
 taken as to the importance of dwelling 
on such considerations in a treatise on 
dynamics1 there can be no doubt that the
 return to ,newton was on the whole a   
salutary change of the manner of        
teaching the subject4  ,the treatment of
 force in the first and second laws of  
motion is frankly causal4 ,force is     
there the cause of rate of change of    
momentum2 and this view ,professor ,tait
 in his own writings has always         
combated1 it must be admitted1 in a very
 cogent manner4 ,according to him1 force
 is merely rate of change of momentum4  
,hence the forces in equations of motion
 are only expressions1 the values of    
which as rates of change of momentum1   
are to be made explicit by the solution 

of such equations in terms of known     
quantities4 ,and there does not seem to 
be any logical escape from this         
conclusion1 though1 except as a way of  
speaking1 the reference to cause        
disappears4  ,the discussion of the     
third law of motion is particularly     
valuable1 for1 as is well known1        
attention was therein called to the fact
 that in the last sentences of the      
,scholium which ,newton appended to his 
remarks on the third law1 the rates of  
working of the acting and reacting      
forces between the bodies are equal and 
opposite4 ,thus the whole work done in  
any time by the parts of a system on one
 another is zero1 and the doctrine of   
conservation of energy is virtually     
contained in ,newton's statement4 ,the  
only point in which the theory was not  
complete so far as ordinary dynamical   
actions are concerned1 was in regard to 
work done against friction1 for which1  
when heat was left out of account1 there
 was no visible equivalent4 ,newton's   

statement of the equality of what       
,thomson and ,tait called activity and  
counter-activity is1 however1 perfectly 
absolute4 ,in the completion of the     
theory of energy on the side of the     
conversion of heat into work1 ,thomson1 
as we have seen1 took a very prominent  
part4  ,after the introduction of the   
dynamical laws the most interesting part
 of this chapter is the elaborate       
discussion which it contains of the     
,lagrangian equations of motion1 of the 
principle of ,least ,action1 with the   
large number of extremely important     
applications of these theories4 ,the    
originality and suggestiveness of this  
part of the book1 taken alone1 would    
entitle it to rank with the great       
classics--the ,mcanique ,cleste1 the    
,mcanique ,analytique1 and the memoirs  
of ,jacobi and ,hamilton--all of which  
were an outcome of the ,principia1 and  
from which1 with the ,principia1 the    
authors of the ,natural ,philosophy drew
 their inspiration4  ,it is perhaps the 

case1 as ,professor ,tait himself       
suggested1 that no one has yet arisen   
who can bend to the fullest extent the  
bow which ,hamilton fashioned2 but when 
this ,ulysses appears it will be found  
that his strength and skill have been   
nurtured by the study of the ,natural   
,philosophy4 ,lagrange's equations are  
now1 thanks to the physical reality     
which the expositions and examples of   
,thomson and ,tait have given to        
generalised forces1 coordinates1 and    
velocities1 applied to all kinds of     
systems which formerly seemed to be     
outside the range of dynamical          
treatment4 ,as ,maxwell put it1 ,the    
credit of breaking up the monopoly of   
the great masters of the spell1 and     
making all their charms familiar in our 
ears as household words1 belongs in     
great measure to ,thomson and ,tait4    
,the two northern wizards were the first
 who1 without compunction or dread1     
uttered in their mother tongue the true 
and proper names of those dynamical     

concepts1 which the magicians of old    
were wont to invoke only by the aid of  
muttered symbols and inarticulate       
equations4 ,and now the feeblest among  
us can repeat the words of power1 and   
take part in dynamical discussions which
 a few years ago we should have left to 
our betters4  ,a very remarkable feature
 in this discussion is the use made of  
the idea of ignoration of coordinates4  
,the variables made use of in the       
,lagrangian equations must be such as to
 enable the positions of the parts of   
the system which determine the motion to
 be expressed for any instant of time4  
,these parts1 by their displacements1   
control those of the other parts1       
through the connections of the system4  
,they are called the independent        
coordinates1 and sometimes the degrees  
of freedom1 of the system4 ,into the    
expressions of the kinetic and potential
 energies1 from which by a formal       
process the equations of motion1 as many
 in number as there are degrees of      

freedom1 are derived1 the value of these
 variables and of the corresponding     
velocities enter in the general case4   
,but in certain cases some of the       
variables are represented by the        
corresponding velocities only1 and the  
variables themselves do not appear in   
the equations of motion4 ,for example1  
when fly-wheels form part of the system1
 and are connected with the rest of the 
system only by their bearings1 the angle
 through which the wheel has turned from
 any epoch of time is of no consequence1
 the only thing which affects the energy
 of the system is the angular velocity  
or angular momentum of the wheel4 ,the  
system is said by ,thomson and ,tait in 
such a case to be under gyrostatic      
domination4 7,see ,gyrostatic ,action1  
p4 #bad below47  ,moreover1 since the   
force which is the rate of growth of the
 momentum corresponding to any          
coordinate is numerically the rate of   
variation with that coordinate of the   
difference of the kinetic and potential 

energies1 every force is zero for which 
the coordinate does not appear2 and     
therefore the corresponding momentum is 
constant4 ,but that momentum is         
expressed by means of the values of     
other coordinates which do appear and   
their velocities1 with the velocities   
for the absent coordinates2 and as many 
equations are furnished by the constant 
values of such momenta as there are     
coordinates absent4 ,the corresponding  
velocities can be determined from these 
equations in terms of the constant      
momenta and the coordinates which appear
 and their velocities4 ,the values so   
found1 substituted in the expressions   
for the kinetic and potential energies1 
remove from these expressions every     
reference to the absent coordinates4    
,then from the new expression for the   
kinetic energy 7in which a function of  
the constant momenta now appears1 and is
 taken as an addition to the potential  
energy7 the equations of motion are     
formed for the coordinates actually     

present1 and these are sufficient to    
determine the motion4 ,the other        
coordinates are thus in a certain sense 
ignored1 and the method is called that  
of ignoration of coordinates4  ,theorems
 of action of great importance for a    
general theory of optics conclude this  
chapter2 but of these it is impossible  
to give here any account1 without a     
discussion of technicalities beyond the 
reading of ordinary students of         
dynamics4  ,in an ,appendix to ,part ,i 
an account is given of ,continuous      
,calculating ,machines4 ,ordinary       
calculating machines1 such as the       
arithmometer of ,thomas of ,colmar1     
carry out calculations and exhibit the  
result as a row of figures4 ,but the    
machines here described are of a        
different character3 they exhibit their 
results by values of a continuously     
varying quantity4 ,the first is one for 
predicting the height of the tides for  
future time1 at any port for which data 
have been already obtained regarding    

tidal heights1 by means of a            
self-registering tide-gauge4 ,two of    
these were made according to the ideas  
set forth in this ,appendix2 one is in  
the ,south ,kensington ,museum1 the     
other is at the ,national ,physical     
,laboratory at ,bushy ,house1 where it  
is used mainly for drawing on paper     
curves of future tidal heights1 for     
ports in the ,indian ,ocean4 ,from these
 curves tide-tables are compiled1 and   
issued for the use of mariners and      
others4  ,another machine described in  
this ,appendix was designed for the     
mechanical solution of simultaneous     
linear equations4 ,it is impossible to  
explain here the interesting arrangement
 of six frames1 carrying as many        
pulleys1 adjustable along slides 7for   
the solution of equations involving six 
unknown quantities71 which ,thomson     
constructed1 and which is now in the    
,natural ,philosophy ,department at     
,glasgow4 ,the idea of arranging the    
first practical machine for this number 

of variables1 was that it might be used 
for the calculation of the corrections  
on values already found for the six     
elements of a comet or asteroid4 ,the   
machine was made1 but some mechanical   
difficulties arose in applying it1 and  
the experiments with it were not at the 
time persevered with4 ,very possibly1   
however1 it may yet be brought into use4
  ,illustration3 ,,fig4 #ad4  ,but the  
most wonderful of these mechanical      
arrangements is the machine for         
analysing the curves drawn by a         
self-registering tide gauge1 so as to   
exhibit the constants of the harmonic   
curves1 and thus enable the prediction  
of tidal heights to be carried out      
either by the tide-predicting machine1  
or by calculation4 ,one day in #ahgf1   
,thomson remarked to his brother1 ,james
 ,thomson1 then ,professor of           
,engineering at ,glasgow1 that all he   
required for the construction of a tidal
 analyser was a form of integrating     
machine more satisfactory for his       

purpose than the usual type of          
integrator employed by surveyors and    
naval architects4 ,james ,thomson at    
once replied that he had invented1 a    
long time before1 what he called a      
disk-globe-cylinder-integrator4 ,this   
consisted of a brass disk1 with its     
plane inclined to the horizontal1 which 
could be turned about its axis by a     
wheel gearing in teeth on the edge of   
the disk1 and driven by the operator in 
a manner which will presently appear4   
,parallel and close to the disk1 but not
 touching it1 was placed a horizontal   
cylinder of brass1 about #b inches in   
diameter 7called the registering        
cylinder71 and between the disk and this
 cylinder was laid a metal ball about #b
 inches in diameter4 ,when the disk was 
kept at rest1 and the ball was rolled   
along between the cylinder and disk1 the
 trace of its rolling on the latter was 
a straight horizontal line passing      
through the centre4 ,supposing then that
 the point of contact of the ball with  

the disk was on one side1 at a distance 
from the centre1 and that the disk was  
then turned1 the ball was by the        
friction between it and the disk made to
 roll1 and so to turn the cylinder4 ,the
 angular velocity of rolling1 and       
therefore the angular velocity of the   
cylinder1 was proportional to the speed 
of the part of the disk in contact with 
it1 that is1 to y4 ,it was also         
proportional to the speed of turning of 
the disk4  ,the mode by which this      
machine effects an integration will now 
be evident4 ,imagine the area to be     
found to lie between a curve and a      
straight datum line1 drawn on a band of 
paper4 ,this is stretched on a large    
cylinder1 with the datum line round the 
cylinder4 ,we call this the             
paper-cylinder4 ,the distances of the   
different points of the curve from the  
datum line are values of y4 ,a          
horizontal bar parallel to the cylinder 
carries a fork at one end and a         
projecting style at the other4 ,the     

globe just fits between the prongs of   
the fork1 and when the bar is moved in  
the direction of its length carries the 
ball along the disk and cylinder4 ,when 
the style at the other end is on the    
datum line1 the centre of the ball is at
 the centre of the disk1 and the turning
 of the disk does not turn the cylinder4
 ,when the bar is displaced in the line 
of its own length to bring the style    
from the datum line to a point on the   
curve1 the ball is displaced a distance 
y1 and there is a corresponding turning 
of the cylinder by the action of the    
ball4 ,in the use of the instrument the 
paper-cylinder is turned by the operator
 while the style is kept on the curve1  
and the disk is turned by the gearing   
already referred to1 which is driven by 
a shaft geared with that of the         
paper-cylinder4 ,thus the displacement  
of the ball is always y1 the ordinate of
 the curve1 and for any displacement dx 
along the datum line1 the registering   
cylinder is turned through an angle     

proportional to ydx4 ,thus any finite   
angle turned through is proportional to 
the integral of ydx for the             
corresponding part of the curve3 a scale
 round one end of the registering       
cylinder gives that angle4 ,thomson     
immediately perceived that this         
extremely ingenious integrating machine 
was just what he required for his       
purpose4 ,the curve of tidal heights    
drawn 7on a reduced scale1 of course7 by
 a tide-gauge1 is really the resultant  
of a large number of simple curves1     
represented by a series of harmonic     
terms1 the coefficients of which are    
certain integrals4 ,the problem is the  
evaluation of these integrals2 and the  
method usually employed is to obtain    
them by measurement of ordinates of the 
curve and an elaborate process of       
calculation4 ,but one of them is simply 
the integral area between the curve and 
the datum line corresponding to the mean
 water level1 and the others are the    
integrals of quantities of the type y   

sin nx4dx1 where y is the ordinate of   
the curve1 and n a number inversely     
proportional to the period of the tidal 
constituent represented by the term4    
,all that was necessary1 in order to    
give the integral of a term y sin nx4dx1
 was to make the disk oscillate about   
its axis as the paper-cylinder was      
turned through an angle proportional to 
x4 ,thus one disk1 globe1 and cylinder  
was arranged exactly as has been        
described for the integral of ydx1 and  
with this as many others as there were  
harmonic terms to be evaluated from the 
curve were combined as follows4 ,the    
disks were placed all in one plane with 
their centres all on one horizontal     
line1 and the cylinders with their axes 
also in line1 and a single sliding bar1 
with a fork for each globe1 gave in each
 case the displacement y from the centre
 of the disk4  ,the requisite different 
speeds of oscillation were given to the 
disks by shafts geared with the         
paper-cylinder1 by trains of wheels cut 

with the proper number of teeth for the 
speed required4  ,thus the angles turned
 through by the registering cylinders   
when a curve on the paper-cylinder was  
passed under the style were proportional
 to the integrals required1 and it was  
only necessary to calibrate the         
graduation of the scales of these       
cylinders by means of known curves to   
obtain the integrals in proper units4   
,one of these machines1 which analyses  
four harmonic constituents1 is in the   
,natural ,philosophy ,department at     
,glasgow2 a much larger machine1 to     
analyse a tidal curve containing five   
pairs of harmonic terms1 or eleven      
constituents in all1 was made for the   
,british ,association ,committee on     
,tidal ,observations1 and is probably   
now in the ,south ,kensington ,museum4  
,but still more remarkable applications 
which ,thomson made of his brother's    
integrating machine were to the         
mechanical integration of linear        
differential equations1 with variable   

coefficients1 to the integration of the 
general linear differential equation of 
any order1 and1 finally1 to the         
integration of any differential equation
 of any order4  ,these applications were
 all made in a few days1 almost in a few
 hours1 after ,james ,thomson first     
described the elementary machine1 and   
papers containing descriptions of the   
combinations required were at once      
dictated by ,thomson to his secretary1  
and despatched for publication4 ,very   
possibly he had thought out the         
applications to some extent before2 but 
it is unlikely that he had done so in   
detail4 ,but1 even if it were so1 the   
connection of a series of machines by   
the single controlling bar1 and the     
production of the oscillations of the   
disks1 all controlled1 as they were1 by 
the motion of a simple point along the  
curve1 so as to give the required       
,fourier coefficients1 were almost      
instantaneous1 and afford an example of 
invention amounting to inspiration4     

,there should be noticed here also the  
geometrical slide for use in            
safety-valves1 cathetometers and other  
instruments1 and the hole-slot-and-plane
 mode of so supporting an instrument now
 used in all laboratories4 ,these were  
,thomson's inventions1 and their        
importance is insisted on in the        
,natural ,philosophy4   ,in ,part ,,ii1 
the principal subjects treated are      
attractions1 elasticity1 such great     
hydrostatical examples as the           
equilibrium theory of the tides and the 
equilibrium of rotating liquid          
spheroids1 and such problems of         
astronomical and terrestrial dynamics as
 the distribution of matter in the      
earth1 with the bearing on this subject 
of the precession of the equinoxes1     
tidal friction1 the earth's rigidity1   
the effects of elastic tides1 the       
secular cooling of the earth1 the age of
 the earth1 and the age of the sun's    
heat4 ,of these1 with the exception of  
the age of the earth1 we shall not      

attempt to give any account4 ,the       
importance of the original contributions
 to elasticity contained in the book is 
indicated by the large space devoted to 
the ,natural ,philosophy in ,professor  
,karl ,pearson's continuation of        
,todhunter's ,history of ,elasticity4   
,the heavy task of editing ,part ,,ii   
was performed mainly by ,sir ,george    
,darwin1 who made many notable additions
 from his own researches to the matter  
contained in the first edition4  ,in the
 next chapter an attempt will be made to
 present ,thomson's views on the subject
 of the age of the earth4 ,these1 when  
they were published1 attracted much     
attention1 and received a good deal of  
hostile criticism from geologists and   
biologists1 whose processes they were   
deemed to restrict to an entirely       
inadequate period of time4              
,,gyrostatic ,,action  ,thomson in his  
lectures and otherwise gave a great deal
 of attention to the motion of          
gyrostats1 and to the effect of the     

inclusion of gyrostats in a system on   
its properties4 ,reference has been made
 to the treatment of gyrostatic         
domination in ,thomson and ,tait4 ,a    
gyrostat consists of a disk or wheel    
with a massive rim1 which revolves      
within a case or framework1 by which the
 whole arrangement can be moved about1  
or supported1 without interfering with  
the wheel4 ,the ordinary toy consisting 
of wheel with a massive rim1 and a light
 frame1 is an example4 ,but much larger 
and more carefully made instruments1 in 
which the wheel is entirely enclosed1   
give the most interesting experiments4  
,the body seems to have its properties  
entirely altered by the rotation of the 
wheel1 and of course the case prevents  
any outward change from being visible4  
,illustration3 ,,fig4 #ae4  ,figure #ae 
shows one form of gyrostat mounted on a 
horizontal frame1 held in the hands of  
an experimenter4 ,the axis of the       
fly-wheel is vertical within the tubular
 part of the case2 the fly-wheel is     

within the part on which is engraved an 
arrow-head to show the direction of     
rotation4 ,round the case in the plane  
of the wheel is a projecting rim        
sharpened to an edge1 on which the      
gyrostat can be supported in other      
experiments4 ,to the rim are screwed two
 projecting pivots1 which can turn in   
bearings on the two sides of the frame  
as shown4 ,the centre of mass of the    
wheel is on the level of these pivots1  
so that the instrument will remain with 
either end of the axis up4  ,if the     
fly-wheel be not in rotation1 the       
experimenter can carry the arrangement  
about1 and the fly-wheel and case move  
with it as if the gyrostat were merely  
an ordinary rigid body4 ,but now remove 
the gyrostat from the frame1 and set the
 wheel in rotation4 ,this is done by an 
endless cord wrapped round a small      
pulley fast on the axle 7to which access
 is obtained by a hole just opposite in 
the case7 and passed also round a larger
 pulley on the shaft of a motor4 ,when  

the motor is started the cord must be   
tightened only very gently at first1 so 
that it slips on the pulley1 otherwise  
the motor would be retarded1 and        
possibly burned by the current4 ,the    
fly-wheel gradually gets up speed1 and  
then the cord can be brought quite tight
 so that no slipping occurs4 ,when the  
speed is great enough the cord is cut   
with a stroke from a sharp knife and    
runs out4  ,the gyrostat is now replaced
 on its pivots in the frame1 with its   
axis vertical1 and moved about as it was
 before4 ,if the experimenter1 holding  
the frame as shown1 turns round in the  
direction of the arrow1 which is that of
 rotation1 nothing happens4 ,if1        
however1 he turns round the other way1  
the gyrostat immediately turns on its   
pivots so as to point the other end of  
the axis up4 ,if the experimenter       
continues his turning motion1 the       
gyrostat is now quiescent3 for it is    
being carried round now in the direction
 of rotation4 ,thus1 with no            

gravitational stability at all 7since   
the centre is on a level with the       
pivots7 the gyrostat is in stable       
equilibrium when carried round in the   
direction of rotation1 but is in        
unstable equilibrium when carried round 
the opposite way4  ,thus1 if the        
observer knew nothing of the rotation of
 the fly-wheel1 and could see and feel  
only the outside of the case1 the       
behaviour of the instrument might well  
appear very astonishing4  ,this is a    
case of what ,thomson and ,tait call    
gyrostatic domination1 which is treated 
very fully in their ,sections #cde 7vi7 
to #cde 7xxviii7 of ,part ,i4 ,it may be
 remarked here that this case of motion 
may be easily treated mathematically in 
an exceedingly elementary manner1 and   
the instability of the one case1 and the
 stability of the other1 made clear to  
the beginner who has only a notion of   
the composition of angular momenta about
 different axes4  ,a year or two ago it 
was suggested by ,professor ,pickering1 

of ,harvard1 that the fact that the     
outermost satellite of ,saturn revolves 
in the direction opposite to the        
planet's rotation1 may be due to the    
fact that originally ,saturn rotated in 
the direction of the motion of this     
moon1 but inasmuch as his motion round  
the sun was opposite in direction to his
 rotation1 he was turned1 so to speak1  
upside down1 like the gyrostat6 ,the    
other satellites1 it is suggested1 were 
thrown off later1 as their revolution is
 direct4 ,professor ,pickering refers to
 an experiment 7similar to that         
described above7 which he gives as new4 
,thomson had shown this experiment for  
many years1 as an example of the general
 discussion in ,thomson and ,tait1 and  
its theory had already been explicitly  
published4#ba  ,many other experiments  
with gyrostats used to be shown by      
,thomson to visitors4 ,many of these are
 indicated in ,thomson and ,tait4 ,the  
earth's precessional motion is a        
gyrostatic effect due to the            

differential attraction of the sun1     
which tends to bring the plane of the   
equator into coincidence with the       
ecliptic1 and so alters the direction of
 the axis of rotation4 ,old students    
will remember the balanced globe--with  
inclined material axis rolling round a  
horizontal ring--by which the kinematics
 of the motion could be studied1 and the
 displacement of the equinoxes on the   
ecliptic traced4  ,illustration3 ,,fig4 
#af4  ,another example of the gyrostatic
 domination discussed in ,thomson and   
,tait is given in the very remarkable   
address entitled ,a ,kinetic ,theory of 
,matter1 which ,sir ,william ,thomson   
delivered to ,section ,a of the ,british
 ,association at ,montreal1 in #ahhd4   
,figure #af shows an ordinary double    
coach spring1 the upper and lower       
members of which carry two hooked rods  
as shown4 ,if the upper hook is attached
 to a fixed support1 and a weight is    
hung on the lower1 the spring will be   
drawn out1 and the arrangement will be  

in equilibrium under a certain          
elongation4 ,if the weight be pulled    
down further and then left to itself1 it
 will vibrate up and down in a period   
depending upon the equilibrium          
elongation produced by the weight4 ,the 
same thing will happen if a spiral      
spring be substituted for the coach     
spring4 ,a spherical case1 through which
 the hooked rods pass freely1 hides the 
internal parts from view4               
,illustration3 ,,fig4 #ag4  ,figure #ag 
shows two hooked rods1 as in the former 
case1 attached by swivels to two        
opposite corners of a frame formed of   
four rods jointed together at their     
ends4 ,each of these is divided in the  
middle for the insertion of a gyrostat1 
the axis of which is pivoted on the     
adjacent ends of the two halves of the  
rod4 ,a spherical case1 indicated by the
 circle1 again hides the internal       
arrangement from inspection1 but permits
 the hooked rods to move freely up and  
down4 ,the swivels allow the frame1     

gyrostats and all1 to be turned about   
the line of the hooks4  ,if now the     
gyrostats be not in rotation1 the frame 
will be perfectly limp1 and will not in 
the least resist pull applied by a      
weight4 ,but if the gyrostats be rotated
 in the directions shown by the circles1
 with arrow-heads drawn round the rods1 
there will be angular momentum of the   
whole system about the line joining the 
hooks1 and if a weight or a force be    
applied to pull out the frame along that
 line1 the pull will be resisted just as
 it was in the other case by the spring4
 ,moreover1 equilibrium will be obtained
 with an elongation proportional to the 
weight hung on1 and small oscillations  
will be performed just as if there were 
a spring in the interior instead of the 
gyrostats4  ,according as the frame is  
pulled out1 or shortened1 the angular   
momentum of the gyrostats about the line
 joining the hooks is increased or      
diminished1 and the frame1 carrying the 
gyrostats with it1 turns about the      

swivels in one direction or the other1  
at the rate necessary to maintain the   
angular momentum at a constant value4   
,but this will not be perceived from    
without4  ,the rotation of the          
fly-wheels thus gives to the otherwise  
limp frame the elasticity which the     
spring possesses2 without dissection of 
the model the difference cannot be      
perceived4 ,this illustrates ,thomson's 
idea that the elasticity of matter may  
be due to motion of molecules or groups 
of molecules of the body1 imbedded in a 
connecting framework1 deformed by       
applied forces as in this model1 and    
producing displacements which are       
resisted in consequence of the motion4  
,and here may be mentioned also         
,thomson's explanation of the           
phenomenon1 discovered by ,faraday1 of  
the rotation of the plane of a beam of  
polarised light which is passed along   
the lines of force of a magnetic field4 
,this rotation is distinct altogether   
from that which is produced when        

polarised light is passed along a tube  
filled with a solution of sugar or      
tartaric acid4 ,if the ray be reflected 
after passage1 and made to retraverse   
the medium1 the rotation is annulled in 
the latter case1 it is doubled in the   
former4 ,this led ,thomson to the view  
that in sugar1 tartaric acid1 quartz1   
etc41 the turning is due to the         
structure of the substance1 and in the  
magnetic field to rotation already      
existing in the medium4 ,he used to say 
that a very large number of minute      
spiral cavities all in the same         
direction1 and all right-handed or all  
left-handed1 in the sugar or quartz1    
would give the effect2 on the other     
hand1 the magnetic phenomenon could only
 be produced by some arrangement        
analogous to a very large number of     
tops1 or gyrostats1 imbedded in the     
medium with their axes all in one       
direction 7or preponderatingly so7 and  
all turning the same way4 ,the rotation 
of these tops or gyrostats ,thomson     

supposed to be caused by the magnetic   
field1 and to be essentially that which 
constitutes the magnetisation of the    
medium4  ,let the frame of the          
gyrostatic spring-balance described     
above1 turn round the line joining the  
hooks so as to exactly compensate1 by   
turning in the opposite direction1 the  
angular momentum about that line given  
by the fly-wheels2 then the arrangement 
will have no angular momentum on the    
whole2 and a large number of such       
balances1 all very minute and hooked    
together1 will form a substance without 
angular momentum in any part4 ,but now  
by the equivalent of a magnetic force   
along the lines of the hooks1 let a     
different angular turning of the frames 
be produced2 the medium will possess a  
specific angular momentum in every part4
 ,if a wave of transverse vibrations    
which are parallel to one direction     
7that is1 if the wave be                
plane-polarised7 enter the medium in the
 direction of the axes of the frames1   

the direction of vibration will be      
turned as the wave proceeds1 that is1   
the plane of polarisation will be turned
 round4  ,more recent research has shown
 an effect of a magnetic field on the   
spectrum of light produced in the field1
 and viewed with a spectroscope in a    
direction at right angles to the        
field--the ,zeeman effect1 as it is     
called--and the explanation of this     
effect by equations of moving electric  
charges1 which are essentially          
gyrostatic equations1 is suggestive of  
an analogy or correspondence between the
 systems of moving electrons which      
constitute these charges1 and some such 
gyrostatic molecules as ,thomson        
imagined4 ,it has been pointed out that 
the ,zeeman effect1 in its simple forms 
at least1 can be exactly imitated by the
 motion of an ordinary pendulum having a
 gyrostat in its bob1 with its axis     
directed along the suspension rod4#bb   
,,electrostatics ,,and ,,magnetism  ,in 
the ten years from #ahfc to #ahgc       

,thomson was extremely busy with        
literary work4 ,in #ahgb1 five years    
after the publication of the treatise on
 ,natural ,philosophy1 and just before  
the appearance of the ,elements1        
,messrs4 ,macmillan  ,co4 published for 
him a collection of memoirs entitled    
,reprint of ,papers on ,electrostatics  
and ,magnetism4 ,the volume contains    
#eif pages1 and the subjects dealt with 
range from the ,uniform ,motion of ,heat
 and its ,connection with the           
,mathematical ,theory of ,electricity   
7the paper already described in ,chapter
 ,,ii above7 and the discussion of      
,electrometers and ,electrostatic       
,measuring ,instruments1 to a complete  
mathematical theory of magnetism4 ,the  
subject of electrostatics led naturally 
to the consideration of electrical      
measuring instruments as they existed   
forty years ago 7about #ahfg71 and their
 replacement by others1 the indications 
of which from day to day should be      
directly comparable1 and capable of     

being interpreted in absolute units4    
,down to that time people had been      
obliged to content themselves with      
gold-leaf electroscopes1 and indeed it  
was impossible for accurate measuring   
instruments to be invented until a      
system of absolute units had been       
completely worked out4 ,the task of     
fixing upon definitions of units and of 
realising them in suitable standards had
 been begun by the ,british             
,association1 and it was as part of the 
,report of that ,committee to the       
,dundee ,meeting in #ahfg that          
,thomson's paper on ,electrometers first
 appeared4  ,it was there pointed out   
that an electrometer is essentially an  
instrument for measuring differences of 
electric potential between conductors1  
by means of effects of electrostatic    
force4 ,such a difference is what a     
gold-leaf electroscope indicates for its
 gold leaves and the walls surrounding  
the air-space in which they are         
suspended4 ,as electroscopes used to be 

constructed1 these walls were made of   
glass imperfectly covered1 if at all1 by
 conducting material1 and the           
electroscope was quite indefinite and   
uncertain in its action4 ,the instrument
 was also1 as made1 quite insensitive4  
,recently1 however1 it has been         
rehabilitated in reputation1 and brought
 into use as a very sensitive indicator 
of effects of radio-activity4  ,thomson 
described in this paper six species of  
electrometers of his own devising4 ,the 
best known of these are his quadrant    
electrometer and his attracted-disk     
electrometers4 ,the former is to be     
found in some form or other in every    
laboratory nowadays1 and need not be    
described in detail4 ,the action is of  
two conductors--the two pairs of        
opposite quadrants of a shallow1        
horizontal1 cylindrical box1 made by    
dividing the box into four by two slits 
at right angles--upon an electrified    
slip of aluminium suspended by a        
two-thread suspension within the box1   

with its length along one of the slits4 
,the two pairs of opposite quadrants are
 at the potential difference to be      
measured1 and the slip of aluminium1 or 
needle1 has each end urged round from a 
quadrant at higher potential towards one
 at a lower1 and these actions conspire 
to turn the slip against its tendency to
 return to the position in which the two
 threads are in one plane4 ,thus the    
deflection 7measured by the displacement
 of a reflected ray of light used as    
index7 gives an indication of the amount
 of the potential difference4  ,the     
electrification of the needle was kept  
up by enclosing the quadrantal box      
within an electrified ,leyden jar1 to   
the interior coating of which contact is
 made by a platinum wire1 depending from
 the needle to sulphuric acid contained 
in the jar4 ,the whole apparatus was    
enclosed in a conducting case connected 
to earth4 ,this made its action         
perfectly definite4 ,variations of this 
electrification of the jar were shown by

 an attached attracted-disk             
electrometer1 the principle of which we 
shall merely indicate4  ,the quadrant   
electrometer has now been vastly        
increased in sensibility by the use of a
 single quartz fibre as suspension4 ,by 
the invention of this fibre1 which is   
exceedingly strong and is1 moreover1 so 
definite in its elastic properties that 
it comes back at once exactly to its    
former zero state after twist1 ,mr4 ,c4 
,v4 ,boys has increased the delicacy of 
all kinds of suspended indicators many  
fold4 ,but it ought to be remembered    
that a ,dolezalek electrometer1 with    
some hundred or more times the          
sensibility of the bifilar instrument1  
was only made possible by its           
predecessor4                            
,attracted-disk-electrometers simply    
measure1 either by weighing or by the   
deflection of a spring1 the attractive  
force between two parallel disks at     
different potentials4 ,from the         
determination of this force1 and the    

measurement of the distance between the 
disks 7or better1 of an alteration of   
the distance7 a difference of potentials
 can be determined1 and a unit for it   
obtained1 which is in direct and known  
relation to ordinary dynamical units4   
,thomson's ,absolute ,electrometer was  
designed specially for accurate         
determinations of this kind4 ,another   
form1 called the ,long ,range           
,electrometer1 was devised for the      
measurement of the potentials of the    
charged conductors in electric machines 
and ,leyden jars4  ,accurate            
determinations of the sparking          
resistance between parallel plates      
charged to different potentials in air  
were made by means of                   
attracted-disk-electrometers in the     
course of some important experiments    
described in the ,electrostatics and    
,magnetism4 ,these results have been    
much referred to in later researches4   
,a small attracted-disk-electrometer was
 used as indicated above to keep a watch

 on the electrification of the ,leyden  
jar of the quadrant instrument1 and a   
small induction machine was added1 by   
turning which the operator could make   
good any loss of charge of the jar4     
,this electrical machine was an example 
of an apparatus on precisely the same   
principle as the ,voss or ,wimshurst    
machines of the present day4 ,in it by a
 set of moving carriers1 influenced by  
conductors1 the charges of the latter   
were increased according to a compound  
interest principle only interfered with 
by leakage to the air or by the         
supports4 ,several forms of this        
machine1 on the same principle1 were    
constructed by ,thomson1 and described  
in #ahfh2 but he afterwards found that  
he had been anticipated by ,c4 ,f4      
,varley in #ahfj4 ,still later it was   
discovered that a similar instrument had
 been made a century before by          
,nicholson1 and called by him the       
,revolving ,doubler4  ,the experiments  
which ,thomson made on atmospheric      

electricity at the old ,college tower1  
and by means of portable electrometers  
in ,arran and elsewhere1 can only be    
mentioned4 ,they led no doubt to some   
improvements on electrometers which he  
made1 the method of bringing the nozzle 
of a water-dropper1 or a point on a     
portable electrometer to the potential  
of the air1 by the inductive action on a
 stream of water-drops in the one case1 
or the particles of smoke from a burning
 match in the other4 ,he invented a     
self-acting machine1 worked by a stream 
of water-drops1 for accumulating        
electric charges1 on the principle of   
the revolving doubler4 ,it was this     
apparently that led to the machines with
 revolving carriers1 to which reference 
has been made above4  ,the mathematical 
theory of magnetism which ,thomson gave 
in #ahdi1 in the ,phil4 ,trans4 ,r4,s41 
was1 when completed by various later    
papers1 a systematic discussion of the  
whole subject1 including                
electromagnetism and diamagnetism4 ,to a

 large extent the ground covered by the 
#ahdi paper had been traversed before by
 ,poisson1 and partially by ,murphy and 
,green2 but ,thomson stated that one    
chief object of his memoir was to       
formally construct the theory without   
reference to the two magnetic fluids1 by
 means of which the facts of experiment 
and conclusions of theory had so far    
been expressed4 ,he found it1 however1  
convenient to introduce the idea of     
positive and negative magnetic matter   
7attracting and repelling as do charges 
of positive and negative electricity71  
which are to be regarded as always      
present in equal amounts1 not only in a 
magnet as a whole1 but in every portion 
of a magnet2 and at first sight this    
might appear like a return to the       
magnetic fluids4 ,but it amounts on the 
whole rather to a conception of a magnet
 as a conglomeration of doublets of     
magnetic matter 7that is1 very close1   
equal and inseparable charges of the two
 kinds of matter71 the arrangement of   

which can be changed by the action of   
magnetic force4 ,this idea is set forth 
now in all the books on magnetism and   
electricity4 ,there can be no doubt that
 the systematic presentment of the      
subject by ,thomson1 and the theorems   
and ideas of magnetic force and magnetic
 permeability by which he rendered the  
clear1 and therefore mathematical1      
notions of ,faraday explicitly          
quantitative1 had much influence in     
furthering the progress of electrical   
science1 and so leading on the one hand 
to the electromagnetic theories of      
,maxwell1 and on the other to modern    
research on the magnetic properties of  
iron1 and to the correct ideas which now
 prevail as to construction of          
dynamo-electric machines and motors4    
,,chapter ,,xii  ,,the ,,age ,,of ,,the 
,,earth   ,from his student days        
throughout his life1 ,lord ,kelvin took 
a keen interest in geological questions4
 ,he was always an active member of the 
,geological ,society of ,glasgow1 and   

was its president for twenty-one years  
7#ahgb-ahic74 ,the distribution of heat 
in the substance of the earth was the   
subject of his inaugural dissertation as
 ,professor of ,natural ,philosophy2 and
 previously1 as a student1 he had       
written an essay on ,the ,figure of the 
,earth1 for which he had been awarded a 
,university ,gold ,medal4 ,he never     
ceased to ponder over the problems of   
terrestrial physics1 and he wrote much  
on the subject4 ,his papers are to be   
found as ,appendices to ,thomson and    
,tait's ,natural ,philosophy1 and in    
vol4 ii of his ,popular ,lectures and   
,addresses1 which is devoted to geology 
and general physics4  ,his conclusions  
regarding the age of the earth have been
 referred to in the last chapter4 ,the  
first allusion to the subject was       
contained 7see p4 #fe above7 in his     
inaugural dissertation ,de ,caloris     
distributione in ,terr ,corpus2 but he  
returned to it again in a communication 
made to the ,royal ,society of          

,edinburgh in ,december1 #ahfe1 and     
entitled ,the ,doctrine of ,uniformity  
in ,geology briefly refuted4 ,on        
,february #bg1 #ahfh1 he delivered to   
the ,geological ,society of ,glasgow an 
address entitled ,on ,geological ,time1 
in which the necessity for limiting     
geological and other changes to an      
almost infinitesimal fraction of the    
vast periods at that time demanded was  
insisted on1 and which gave rise to much
 discussion4  ,the address began with a 
protest against the old uniformitarian  
view of geological changes as expressed 
by ,playfair in his ,illustrations of   
the ,huttonian ,theory4 ,the first      
objection taken to the idea that in the 
continuation of the different species of
 animals and vegetables that inhabit the
 earth1 we discern neither a beginning  
nor an end2 in the planetary motions    
where geometry has carried the eye so   
far1 both into the future and the past1 
we discover no mark either of the       
commencement or the termination of the  

present order is1 that the stability of 
the motions of the heavenly bodies1 to  
which reference is made in this         
statement1 is founded upon what is      
essentially an approximate calculation1 
which leaves out1 by intention1 the     
consideration of frictional resistance4 
,he points out1 for example1 that the   
friction which accompanies the relative 
motion of the waters of the earth and   
the land is attended by the production  
of heat1 and that1 by the doctrine of   
the conservation of energy1 heat cannot 
be produced without a disappearance of  
an equivalent quantity of energy1 either
 of motion or of position4 ,the chief   
source of this energy is the earth's    
rotation4 ,since the earth turns under  
the moon and the tidal spheroid--that   
is1 the earth's shape as distorted by   
the heaping up of the waters in the     
tides--remains on the whole stationary  
with respect to the moon1 the solid     
matter of the earth turns under the     
distribution of the water1 held more or 

less fixed by the moon1 as does a       
fly-wheel under a stationary friction   
band round its rim4 ,then just as the   
band held fixed retards the fly-wheel1  
so the earth must be retarded in its    
rotation by this water-brake4 ,in the   
earth's rotation there is a store of    
kinetic energy which1 roughly estimated1
 would not be exhausted in less than ten
 million million years1 although drawn  
upon continuously by friction1 or other 
actions1 at the rate of one million     
horse-power2 so that1 no immediate      
catastrophe1 such as that we should be  
involved in by the stoppage or          
considerable retardation of the spinning
 motion of the earth1 is possible4 ,but 
it was pointed out by ,thomson that the 
best results of astronomical observation
 show that the earth would in one       
hundred years fall behind a perfect     
time-keeper1 with which its rotation    
kept pace at the beginning of the time1 
by about twenty seconds4 ,the tendency  
is to make the earth turn slower1 and   

the moon to increase its distance and   
move more slowly in its orbit1 but with 
a resultant effect towards coincidence  
of the period of the earth's rotation   
with that of revolution of the moon     
round the earth4 ,after this coincidence
 has been attained1 however1 the solar  
tides will tend to make the moon fall in
 towards the earth4  ,if then the earth 
be rotating more and more slowly1 as    
time goes on1 at present1 it must have  
been rotating more rapidly in past time4
 ,a thousand million years ago1 at the  
present rate of retardation1 the earth  
must have been rotating one seventh part
 of its speed faster than it is rotating
 at present1 and this would give for    
centrifugal force at the surface one    
thousand million years ago1 greater than
 the centrifugal force at present1 in   
the ratio of #fd to #di4 ,apparently    
therefore the earth must have solidified
 at a much later date than that epoch1 a
 date when it was rotating much more    
nearly with the angular speed which it  

has now2 otherwise the figure of the    
earth would have deviated much more from
 the spherical form than it actually    
does4 ,on the other hand1 one hundred   
million years ago centrifugal force     
would be only three per cent4 greater   
than it is at present1 and consolidation
 of the earth at that less remote period
 would give a shape to the earth not    
very different from that which it now   
possesses4 ,the argument therefore from 
tidal retardation would cut down the    
time available for geological and       
biological changes to something not much
 more than one hundred million years1   
perhaps to less4  ,a second argument for
 limitation of the time available for   
such processes is derived from the sun's
 heat4 ,the sun cannot be regarded as a 
miraculous body producing its light and 
heat from nothing4 ,changes of the      
constitution of the sun must be         
continually proceeding1 to account for  
its enormous radiation of energy into   
space1 a radiation of which only an     

infinitesimal part is received by the   
bodies of the solar system1 and a still 
more minute portion by the earth4 ,the  
effects of the sun's light and heat on  
the earth show how enormous must be the 
quantity of energy lost from the sun in 
a year4 ,how is this loss of energy to  
be accounted for8 ,what is the physical 
change which gives rise to it8 ,in #ahed
 ,thomson put forward the theory that   
the sun's heat is kept up by the falling
 in of meteors on the sun's surface1 but
 he afterwards saw reason to abandon    
that view4 ,helmholtz had advocated the 
theory that the sun was a body heated by
 the coming together of the matter      
composing it by its mutual attraction1 a
 process which1 although the sun is now 
a continuous mass1 is to be regarded as 
still going on4 ,it is easy to calculate
 the exhaustion of potential energy     
caused by the coming together of the    
matter of the sun from universal        
dispersion through infinite space to a  
sphere of uniform density of the present

 size of the sun4 ,the result is about  
as much energy as would be generated by 
burning seven million million million   
million million tons of coal4 ,the      
amount radiated in each hour is about as
 much as would be generated by burning  
something like nine tons of coal every  
hour on every square yard of the sun's  
surface4 ,it is certain that the sun    
must be still contracting1 and if it    
contracts sufficiently to just make good
 this expenditure by the further        
exhaustion of potential energy involved 
in the closer aggregation of the matter1
 it must diminish in radius in each year
 by as much as #acj feet4  ,the amount  
of energy generated by the falling      
together of the matter of the sun from  
universal diffusion to the dimensions   
which the sun has at present1 is only   
about #ac1jjj1jjj times the amount now  
radiated per annum4 ,in ,thomson's paper
 ,pouillet's estimate of the energy     
radiated per second is used1 and this   
number is raised to #bj1jjj1jjj4 ,taking

 the latter estimate1 the whole         
potential energy exhausted by the       
condensation of the sun's mass to       
uniform density would suffice for only  
#bj1jjj1jjj years' supply4 ,but the sun 
is undoubtedly of much greater density  
in the central parts than near the      
surface1 and so the energy exhausted    
must be much greater than that stated   
above4 ,this will raise the number of   
years provided for4 ,on the other hand1 
a considerable amount of energy would be
 dissipated during the process of       
condensation1 and this would reduce the 
period of radiation estimated4 ,thomson 
suggests that #ej1jjj1jjj1 or           
#ajj1jjj1jjj1 years is a possible       
estimate4  ,it is not unlikely that the 
rate of radiation in past time1 when the
 sun had not nearly condensed to its    
present size1 was so much less than it  
is at present that the period suggested 
above may have to be considerably       
augmented4 ,another source of radiation1
 which seems to be regarded by some     

authorities as a probable1 if not a     
certain1 one1 has been suggested in     
recent years--the presence of           
radio-active substances in the sun4 ,so 
far as we know1 ,lord ,kelvin did not   
admit that this source of radiation was 
worthy of consideration2 but of course1 
granted its existence to an extent      
comparable with the energy derivable    
from condensation of the sun's mass1 the
 age of the sun's heat would have to be 
very greatly extended4 ,these are       
matters1 however1 on which further light
 may be thrown as research in           
radio-activity progresses4 ,lord ,kelvin
 was engaged when seized with his last  
illness in discussing the changes of    
energy in a gaseous1 or partially       
gaseous1 globe1 slowly cooling and      
shrinking in doing so2 and a posthumous 
paper on the subject will shortly be    
published which may possibly contain    
further information on this question of 
solar physics4  ,but ,thomson put       
forward a third argument in the paper on

 ,geological ,time1 which has always    
been regarded as the most important4 ,it
 is derived from the fact1 established  
by abundant observations1 that the      
temperature in the earth's crust        
increases from the surface inwards2 and 
that therefore the earth must be        
continually losing heat by conduction   
from within4 ,if the earth be supposed  
to have been of uniform temperature at  
some period of past time and in a molten
 state1 and certain assumptions as to   
the conductive power and melting point  
of its material be made1 the time of    
cooling until the gradient of           
temperature at the surface acquired its 
present value can be calculated4 ,this  
was done by ,thomson in a paper         
published in the ,transactions1         
,r4,s4,e41 in #ahfb4 ,we propose to give
 here a short sketch of his argument1   
which has excited much interest1 and    
been the cause of some controversy4  ,in
 order to understand this argument1 the 
reader must bear in mind some           

fundamental facts of the flow of heat in
 a solid4 ,let him imagine a slab of any
 uniform material1 say sandstone or     
marble1 the two parallel faces of which 
are continually maintained at two       
different temperatures1 uniform over    
each face4 ,for example1 steam may be   
continually blown against one face1     
while ice-cold water is made to flow    
over the other4 ,heat will flow across  
the slab from the hotter face to the    
colder4 ,it will be found that the rate 
of flow of heat per unit area of face1  
that is per square centimetre1 or per   
square inch1 is proportional to the     
difference of the temperatures in the   
slab at the two faces1 and inversely    
proportional to the thickness of the    
slab4 ,in other words1 it is            
proportional to the fall of temperature 
from one face to the other taken per    
unit of the thickness1 that is1 to the  
gradient of temperature from one face to
 the other4 ,moreover1 comparing the    
flow in one substance with the flow in  

another1 we find it different in        
different substances for the same       
gradient of temperature4 ,thus we get   
finally a flow of heat across unit area 
of the slab which is equal to the       
gradient of temperature multiplied by a 
number which depends on the material3   
that number is called the conductivity  
of the substance4  ,now1 borings made in
 the earth show that the temperature    
increases inwards1 and the same thing is
 shown by the higher temperatures found 
in deeper coal mines4 ,by means of      
thermometers sunk to different depths1  
the rate of increase of temperature with
 depth has been determined4 ,similar    
observations show that the daily and    
annual variations of temperature caused 
by the succession of day and night1 and 
summer and winter1 penetrate to only a  
comparatively small depth below the     
surface--three or four feet in the      
former case1 sixty or seventy in the    
latter4 ,leaving these variations out of
 account1 since the average of their    

effects over a considerable interval of 
time must be nothing1 we have in the    
earth a body at every point of the crust
 of which there is a gradient of        
increasing temperature inwards4 ,the    
amount of this may be taken as one      
degree of ,fahrenheit's scale for every 
#ej feet of descent4 ,this gradient is  
not uniform1 but diminishes at greater  
depths4 ,supposing the material of      
uniform quality as regards              
heat-conducting power1 the mathematical 
theory of a cooling globe of solid      
material 7or of a straight bar which    
does not lose heat from its sides7 gives
 on certain suppositions the gradients  
at different depths4 ,the surface       
gradient of #a ,f4 in #ej feet may be   
taken as holding for #ejjj feet or #fjjj
 feet or more4  ,this gradient of       
diminution of temperature outwards leads
 inevitably to the conclusion that heat 
must be constantly flowing from the     
interior of the earth towards the       
surface4 ,this is as certain as that    

heat flows along a poker1 one end of    
which is in the fire1 from the heated   
end to the other4 ,the heat which       
arrives at the surface of the earth is  
radiated to the atmosphere or carried   
off by convection currents2 there is no 
doubt that it is lost from the earth4   
,thus the earth must be cooling at a    
rate which can be calculated on certain 
assumptions1 and it is possible on these
 assumptions to calculate backwards1 and
 determine the interval of time which   
must have elapsed since the earth was   
just beginning to cool from a molten    
condition1 when of course life cannot   
have existed on its surface1 and those  
geological changes which have effected  
so much can hardly have began4          
,considering a globe of uniform         
material1 and of great radius1 which was
 initially at one temperature1 and at a 
certain instant had its surface suddenly
 brought to1 let us say1 the temperature
 of melting ice1 at which the surface   
was kept ever after1 we can find1 by    

,fourier's mathematical theory of the   
flow of heat1 the gradient of           
temperature at any subsequent time for a
 point on the surface1 or at any        
specified distance within it4 ,for a    
point on the surface this gradient is   
simply proportional to the initial      
uniform temperature1 and inversely      
proportional to the square root of the  
product of the diffusivity of the       
material 7the ratio of the conductivity 
to the specific heat7 by the interval of
 time which has elapsed since the       
cooling was started4 ,taking a foot as  
the unit of length1 and a year as the   
unit of time1 we find the diffusivity of
 the surface strata to be #djj4 ,if we  
take the initial temperature as #gjjj   
degrees ,f4--which is high enough for   
melting rock--and take the interval of  
time which has elapsed as #ajj1jjj1jjj  
years1 we obtain at the surface a       
gradient approximately equal to that    
which now exists4 ,a greater interval of
 time would give a lower gradient1 a    

smaller interval would give a higher    
gradient than that which exists at      
present4 ,a lower initial temperature   
would require a smaller interval of     
time1 a higher initial temperature a    
longer interval for the present         
gradient4  ,with the initial temperature
 of #g1jjj degrees ,f41 an interval of  
#d1jjj1jjj years would give a surface   
gradient of #a ,f4 in #aj ft4 ,thus1 on 
the assumption made1 the surface        
gradient of temperature has diminished  
from #aaj to #aej in about #if1jjj1jjj  
years4 ,after #aj1jjj years from the    
beginning of the cooling the gradient of
 temperature would be #b ,f4 per foot4  
,but1 as ,thomson showed1 such a large  
gradient would not lead to any sensible 
augmentation of the surface temperature1
 for the radiation from earth and       
atmosphere into space would almost      
certainly be so rapid as to prevent     
this4 ,hence he inferred that conducted 
heat1 even at that early period1 could  
not sensibly affect the general climate4

  ,two objections 7apart from the       
assumptions already indicated7 will     
readily occur to any one considering    
this theory1 and these ,thomson answered
 by anticipation4 ,the first is1 that no
 natural action could possibly bring the
 surface of a uniformly heated globe    
instantaneously to a temperature #gjjj  
lower1 and keep it so ever after4 ,in   
reply to this ,thomson urged that a     
large mass of melted rock1 exposed      
freely to our earth and sky1 will1 after
 it once becomes crusted over1 present  
in a few hours1 or a few days1 or at    
most a few weeks1 a surface so cool that
 it can be walked over with impunity4   
,hence1 after #aj1jjj years1 or indeed1 
,i may say1 after a single year1 its    
condition will be sensibly the same as  
if the actual lowering of temperature   
experienced by the surface had been     
produced in an instant1 and maintained  
constant ever after4 ,the other         
objection was1 that the earth was       
probably never a uniformly heated solid 

#gjjj ,f4 above the present surface     
temperature as assumed for the purpose  
of calculation4 ,this ,thomson answers  
by giving reasons for believing that the
 earth1 although once all melted1 or    
melted all round its surface1 did1 in   
all probability1 really become a solid  
at its melting temperature all through1 
or all through the outer layer which has
 been melted2 and not until the         
solidification was thus complete1 or    
nearly so1 did the surface begin to     
cool4  ,thomson was inclined to believe 
that a temperature of #gjjj ,f4 was     
probably too high1 and results of       
experiments on the melting of basalt and
 other rocks led him to prefer a much   
reduced temperature4 ,this1 as has      
already been pointed out1 would give a  
smaller value for the age of the earth4 
,in a letter on the subject published in
 ,nature 7vol4 #ea1 #ahie7 he states    
that he is not led to differ much from  
an estimate of #bd1jjj1jjj years founded
 by ,mr4 ,clarence ,king 7,american     

,journal of ,science1 ,january #ahic7 on
 experiments on the physical properties 
of rocks at high temperatures4  ,it is  
to be observed that the assumptions made
 above that the physical constants of   
the material are constant throughout the
 earth1 and at all temperatures1 are    
confessedly far from the truth4         
,nevertheless ,thomson strongly held    
that the uncertainty of the data can at 
most extend the earth's age to some     
value between #bj1jjj1jjj and           
#bjj1jjj1jjj of years1 and that the     
enormously long periods which were wont 
to be asked for by geologists and       
biologists for the changes of the       
earth's surface and the development of  
its flora and fauna1 cannot possibly be 
conceded4  ,in ,nature for ,january #c1 
#ahie1 ,professor ,john ,perry suggested
 that very possibly the conductivity of 
the material composing the interior of  
the earth was considerably higher than  
that of the surface strata4 ,if this    
were so1 then1 as can be shown without  

difficulty1 the attainment of the       
present gradient would be very greatly  
retarded1 and therefore the age of the  
earth correspondingly increased4 ,the   
question then arose1 and was discussed1 
as to whether the rocks and other       
materials at high temperatures were more
 or less conducting than at low         
temperatures1 and experiments on the    
subject were instituted and carried out4
 ,on the whole1 the evidence seemed to  
show that the conductivity of most      
substances is diminished1 not increased1
 by the rise of temperature1 and so far 
as it went1 therefore1 the evidence was 
against ,professor ,perry's suggestion4 
,on the other hand1 he contended that   
the inside of the earth may be a mass of
 great rigidity1 partly solid and partly
 fluid1 possessing a quasi-conductivity 
which might greatly increase the period 
of cooling4 ,the subject is a difficult 
one both from a mathematical and from   
the physical point of view1 and further 
investigation is necessary1 especially  

of the behaviour of materials under the 
enormous stresses which they undoubtedly
 sustain in the interior of the earth4  
,after the publication of the paper on  
,geological ,time a reply to it was made
 by ,professor ,huxley1 in an address to
 the ,geological ,society of ,london1   
delivered on ,february #ai1 #ahfi4 ,he  
adopted the rle of an advocate retained 
for the defence of geology against what 
seems to have been regarded as an       
unwarranted attack1 made by one who had 
no right to offer an opinion on a       
geological question4 ,for1 after a long 
and eloquent pleading1 he concludes his 
address with the words3 ,my functions1  
as your advocate1 are at an end4 ,i     
speak with more than the sincerity of a 
mere advocate when ,i express the belief
 that the case against us has entirely  
broken down4 ,the cry for reform which  
has been raised from without is         
superfluous1 inasmuch as we have long   
been reforming from within with all     
needful speed2 and the critical         

examination of the grounds upon which   
the very grave charge of opposition to  
the principles of ,natural ,philosophy  
has been brought against us1 rather     
shows that we have exercised a wise     
discrimination in declining to meddle   
with our foundations at the bidding of  
the first passer-by who fancies our     
house is not so well built as it might  
be4 ,to this ,thomson rejoined in an    
address entitled ,of ,geological        
,dynamics1 also delivered to the        
,geological ,society of ,glasgow on     
,april #e1 #ahfi2 and to this1 with     
,professor ,huxley's address1 the reader
 must be referred for the objection1    
brought against ,thomson's arguments1   
and the replies which were immediately  
forthcoming4 ,this is not the place to  
discuss the question1 but reference may 
be made to an interesting paper on the  
subject in the ,glasgow ,herald for     
,february #bb1 #aijh1 by ,professor ,j4 
,w4 ,gregory1 in which the suggestion of
 ,professor ,perry1 of a nearer approach

 to uniformity of temperature in the    
interior of the earth than ,thomson had 
thought possible1 is welcomed as        
possibly extending the interval of time 
available to a period sufficient for all
 purposes4 ,in ,professor ,gregory's    
opinion1 ,lord ,kelvin in one respect   
showed a keener insight than ,huxley1   
who1 referring to possible changes in   
the rate of rotation of the earth1 or in
 the heat given forth from the sun or in
 the cooling of the earth1 declared that
 geologists are ,gallios1 'who care for 
none of these things4' ,an              
ever-increasing school of geologists now
 cares greatly for these questions1 and 
reveres ,lord ,kelvin as one of the     
founders of the geology of the inner    
earth4  ,after all1 the problem is not  
one to be dealt with by the geologist or
 biologist alone1 but to be solved1 so  
far as it can be solved at all1 by a    
consideration of all relevant evidence1 
from whatsoever quarter it may come4 ,it
 will not do in these days for          

scientific men to shut themselves up    
within their special departments and to 
say1 with regard to branches of science 
which deal with other aspects of nature 
and other problems of the past1 present 
and future of that same earth on which  
all dwell and work1 that they care for  
none of these things4 ,this is an echo  
of an old spirit1 not yet dead1 that has
 done much harm to the progress of      
science4 ,the division of science into  
departments is unavoidable1 for         
specialisation is imperative2 but it is 
all the more necessary to remember that 
the divisions set up are more or less   
arbitrary1 and that there are absolutely
 no frontiers to be guarded and         
enforced4 ,chemistry1 physiology1 and   
physics cannot be walled off from one   
another without loss to all2 and geology
 has suffered immensely through its     
having been regarded as essentially a   
branch of natural history1 the devotees 
of which have no concern with           
considerations of natural philosophy4   

,lord ,kelvin's dignified questions were
 unanswerable4 ,who are the occupants of
 'our house1' and who is the            
'passer-by'8 ,is geology not a branch of
 physical science8 ,are investigations1 
experimental and mathematical1 of       
underground temperature not to be       
regarded as an integral part of         
geology8444 ,for myself1 ,i am anxious  
to be regarded by geologists not as a   
mere passer-by1 but as one constantly   
interested in their grand subject1 and  
anxious in any way1 however slight1 to  
assist them in their search for truth4  
,,chapter ,,xiii  ,,british             
,,association ,,committee ,,on          
,,electrical ,,standards   ,when        
,professor ,thomson began his work as a 
teacher in the ,university of ,glasgow1 
there was1 as has already been noticed1 
great vagueness of specification of     
physical quantities4 ,few of the formal 
definitions of units of measurement1 now
 to be found in the pages of every      
elementary text book1 had been framed1  

and there was much confusion of         
quantities essentially distinct1 a      
confusion which is now1 to some extent  
at least1 guarded against by the        
adoption of a definite unit1 with a     
distinctive name for each magnitude to  
be measured4 ,thus rate of working1 or  
activity1 was confused with work done2  
the condition for maximum activity in   
the circuit of a battery or dynamo was  
often quoted as the condition of        
greatest efficiency1 that is of greatest
 economy of energy1 although it was     
exactly that in which half the available
 energy was wasted4  ,partly as a       
consequence of this vagueness of        
specification1 there was a great want of
 knowledge of the values of physical    
constants2 for without exact definitions
 of quantities to be determined1 such   
definitions as would indicate units for 
their measurement1 related to ordinary  
dynamical units according to a          
consistent scheme1 it was impossible to 
devise satisfactory experimental methods

 to do for electricity and magnetism    
what had been done by ,regnault and     
others for heat4  ,the first steps      
towards the construction of a complete  
system of units for the quantitative    
measurement of magnetic and electric    
quantities were taken by ,gauss1 in his 
celebrated paper entitled ,intensitas   
vis magnetic terrestris ad mensuram     
absolutam revocata1 published in #ahcb4 
,in this he showed how magnetic forces  
could be expressed in absolute units1   
and thus be connected with the absolute 
dynamical units which ,gauss1 in the    
same paper1 based on chosen fundamental 
units of length1 mass1 and time4 ,thus  
the modern system of absolute units of  
dynamical quantities1 and its extension 
to magnetism1 are due to the practical  
insight of a great mathematician1 not to
 the experimentalists or practicians of 
the time4  ,methods of measuring        
electric quantities in absolute units   
were described by ,w4 ,weber1 in ,parts 
,,ii and ,,iii of his                   

,elecktrodynamische ,maassbestimmungen1 
published in #aheb4 ,these were great   
steps in advance1 and rendered further  
progress in the science of absolute     
measurement comparatively easy4 ,but    
they remained the only steps taken until
 the ,british ,association ,committee   
began their work4 ,we have already 7pp4 
#gd-gf7 referred to the great importance
 of that work1 not only for practical   
applications but also for the           
advancement of science4 ,but it was not 
a task which struck the imagination or  
excited the wonder of the multitude4    
,for the realisation of standards of    
resistance1 for example1 involved long  
and tedious investigations of the       
effects of impurities on the resistance 
of metals1 and the variation of         
resistance caused by change of          
temperature and lapse of time4 ,then    
alloys had to be sought which would have
 a temperature effect of small amount1  
and which were stable and durable in all
 their properties4  ,the discoveries of 

the experimentalist who finds a new     
element of hitherto undreamed-of        
properties attract world-wide attention1
 and the glory of the achievement is    
deservedly great4 ,but the patient1     
plodding work which gives a universal   
system of units and related standards1  
and which enables a great physical      
subject like electricity and magnetism  
to rise from a mere enumeration of      
qualitative results to a science of the 
most delicate and exact measurement1 and
 to find its practical applications in  
all the affairs of daily life and       
commerce1 is equally deserving of the   
admiration and gratitude of mankind4    
,yet it receives little or no           
recognition4  ,the construction of a    
standard of resistance was the first    
task undertaken by the committee2 but   
other units1 for example of quantity of 
electricity1 intensity of electric field
 and difference of potential1 had also  
to be defined1 and methods of employing 
them in experimental work devised4 ,it  

would be out of place to endeavour to   
discuss these units here1 but some idea 
of the manner in which their definitions
 are founded on dynamical conceptions   
may be obtained from one or two         
examples4 ,therefore we shall describe  
two simple experiments1 which will      
illustrate this dynamical foundation4   
,an account has been given in ,chapter  
,,xi of the series of electrometers     
which ,thomson invented for the         
measurement of differences of electric  
potential4 ,these all act by the        
evaluation in terms of ordinary         
dynamical units of the force urging an  
electrified body from a place of higher 
towards a place of lower potential4     
,some indication of the meaning of      
electrical quantities has been given in 
,chapter ,,iv4 ,difference of electric  
potential between two points in an      
electric field was there defined as the 
dynamical work done in carrying a unit  
of positive electricity against the     
forces of the field from the point of   

lower to the point of higher potential4 
,now by the definition of unit quantity 
of electricity given in electrical      
theory--that quantity which1            
concentrated at a point at unit distance
 from an equal quantity also            
concentrated at a point1 is repelled    
with unit force--we can find1 by the    
simple experiment of hanging two pith   
balls 7or1 better1 two hollow1 gilded   
beads of equal size7 by two fine fibres 
of quartz1 a metre long1 say1           
electrifying the two balls as they hang 
in contact1 and observing the distance  
at which they then hang1 the numerical  
magnitude in absolute units of a charge 
of electricity1 and apply that to       
finding the charge on a large spherical 
conductor and the potential at points in
 its field also in absolute units4 ,if m
 be the mass of a ball1 g gravity in cm4
 sec4 units1 d the distance in cms4 of  
the centres of the balls apart1 and l   
the length in cms4 of a thread1 the     
charge q1 say1 on each ball is easily   

found to be mgd#d7l  d74 ,thus the      
charge is got in absolute               
centimetre-gramme-second units in terms 
of the mass m obtained by ordinary      
weighing1 and l and d obtained by easy  
and exact measurements4  ,if one of the 
balls be now taken away without         
discharging the other1 and the latter be
 placed in the field of a large         
electrified spherical conductor1 the    
fibre will be deflected from the        
vertical by the force on the ball4 ,let 
the two centres be now on the same      
level4 ,that force is got at once from  
the angle of deflection 7which is easily
 observed71 the charge on the ball1 and 
the value of m4 ,the electric           
field-intensity is obtained by dividing 
the value of the force by q4 ,the field 
intensity multiplied by ,d1 the distance
 apart in cms4 of the centres of the    
ball and the conductor1 gives the       
potential at the centre of the ball in  
,c4,g4,s4 units4 ,multiplication again  
by ,d gives the charge on the conductor4

  ,when it made its first ,report in    
#ahfb 7to the meeting at ,cambridge7 the
 committee consisted of ,professors ,a4 
,williamson1 ,c4 ,wheatstone1 ,w4       
,thomson1 ,w4 ,h4 ,miller1 ,dr4 ,a4     
,matthiessen1 and ,mr4 ,f4 ,jenkin4 ,at 
the next meeting1 at ,newcastle1 it had 
been augmented by the addition of       
,messrs4 ,balfour ,stewart1 ,c4 ,w4     
,siemens1 ,professor ,clerk ,maxwell1   
,dr4 ,joule1 ,dr4 ,esselbach1 and ,sir  
,charles ,bright4 ,the duty with which  
the committee had been charged was that 
of constructing a suitable standard of  
resistance4 ,a reference to the account 
given in ,chapter ,x above1 of the      
derivation of what came to be called the
 electromagnetic unit of difference of  
potential1 or electromotive force1 by   
means of a simple magneto-electric      
machine--a disk turning on a uniform    
magnetic field1 or the simple rails and 
slider and magnetic field arrangement   
there described--will show how from this
 unit and the electromagnetic unit of   

current 7there also defined7 the unit of
 resistance is defined4 ,it is the      
resistance of the circuit of slider1    
rails1 and connecting wire1 when with   
this electromagnetic unit of            
electromotive force the unit of current 
is made to flow4  ,this was one clear   
and definite way of defining the unit of
 current1 and of attaining the important
 object of connecting the units in such 
a way that the rate of working in a     
circuit1 or the energy expended in any  
time1 should be expressed at once in    
ordinary dynamical units of activity or 
energy4 ,a considerable number of       
proposals were discussed by the         
committee2 but it was finally determined
 to take the basis here indicated1 and  
to realise a standard of resistance in  
material of constant and durable        
properties1 which should have some      
simple multiple of the unit of          
resistance1 in the system of dynamical  
units based on the centimetre as unit of
 length1 the gramme as unit of mass1 and

 the second as unit of time--the        
so-called ,c4,g4,s4 system4 ,the        
comparison of the different metals and  
alloys available was a most important   
but exceedingly laborious series of     
investigations1 carried out mainly by   
,dr4 ,matthiessen and ,professor        
,williamson4  ,professor ,thomson       
suggested to the committee the          
celebrated method of determining the    
resistance of a circuit by revolving a  
coil1 which formed the main part of the 
circuit about a vertical axis in the    
earth's magnetic field4 ,an account of  
the experiments made with this method is
 contained in the ,report of #ahfc4     
,they were carried out at ,king's       
,college1 ,london1 where ,maxwell was   
then ,professor of ,experimental        
,physics1 by ,maxwell1 ,balfour         
,stewart1 and ,fleeming ,jenkin4 ,the   
theoretical discussion and the          
description of the experiments was      
written by ,maxwell1 the details of the 
apparatus were described by ,jenkin4    

,the principle of the method is         
essentially the same as that of the     
simple magneto-electric machine1 to     
which reference has just been made4 ,two
 parallel coils of wire were wound in   
channels cut round rings of brass1      
which1 however1 were cut across by slots
 filled with vulcanite1 to prevent      
induced currents from circulating in the
 brass4 ,these coils were mounted in a  
vertical position and could be driven as
 a rigid system1 at a constant measured 
speed1 about a vertical axis passing    
through the centre of the system4       
,between the coils at this centre was   
hung1 from a steady support1 a small    
magnetic needle by a single fibre of    
silk2 and a surrounding screen prevented
 the needle and suspension from being   
affected by currents of air4  ,the ends 
of the coil were connected together so  
that the whole revolved as a closed     
circuit about the vertical axis4 ,when  
the coil system was at right angles to  
the magnetic meridian there was a       

magnetic induction through it of amount 
,,ah1 where ,a denotes the effective    
area of the coils1 and ,h the horizontal
 component of the earth's magnetic      
field4 ,by one half-turn the coil was   
reversed with reference to this magnetic
 induction1 and as the coil turned an   
induced current was generated1 which    
depended at any instant on the rate at  
which the magnetic induction was varying
 at the instant1 on the inductive       
electromotive force due to the varying  
of the current in the coil itself1 and  
on the resistance of the circuit4 ,a    
periodic current thus flowed in one     
direction relatively to the coil in one 
half-turn from a position perpendicular 
to the magnetic meridian1 and in the    
opposite direction in the next          
half-turn4 ,but as the position of the  
coil was reversed in every half-turn as 
well as the current in it1 the current  
flowed on the whole in the same average 
direction relatively to the needle1 and 
but for self-induction would have had   

its maximum value always when the plane 
of the coil was in the magnetic         
meridian4  ,the needle was deflected as 
it would have been by a certain average 
current1 and the deflection was opposed 
by the action of the earth's horizontal 
magnetic field ,h4 ,but this was the    
field cut by the coil as it turned1 and 
therefore 7except for a small term      
depending on the turning of the coil in 
the field of the needle7 the value of ,h
 did not appear in the result1 and did  
not require to be known4  ,full details 
of the theory of this method and of the 
experiments carried out to test it will 
be found in various memoirs and         
treatises#bc2 but it must suffice here  
to state that the resistance of the coil
 was determined in this way1 by a large 
series of experiments1 before and after 
every one of which the resistance was   
compared with that of a ,german-silver  
standard4 ,the resistance of this       
standard therefore became known in      
absolute units1 and copies of it1 or    

multiples or sub-multiples of it1 could 
be made4  ,a unit called the ,b4,a4     
unit1 which was intended to contain #aji
 ,c4,g4,s4 electromagnetic units of     
resistance1 was constructed from these  
experiments1 and copies of it were soon 
after to be found in nearly all the     
physical laboratories of the world4     
,resistance boxes were constructed by   
various makers1 in which the coils were 
various multiples of the ,b4,a4 unit1 so
 that any resistance within a certain   
range could be obtained by connecting   
these coils in series 7which was easily 
done by removing short circuiting       
plugs71 and thus the absolute units of  
current electromotive force and         
resistance came into general use4  ,in  
#ahha ,lord ,rayleigh and ,professor    
,schuster carried out a very careful    
repetition of the ,british ,association 
experiments with the same apparatus at  
the ,cavendish ,laboratory1 and obtained
 a somewhat different result4 ,they     
found that the former result was about  

#a4ag per cent4 too small4 ,lord        
,rayleigh next carried out an           
independent set of experiments by the   
same method with improved apparatus1 and
 found that this percentage error must  
be increased to about #a4ce4  ,it may be
 noticed here that the simple disk      
machine1 of ,thomson's illustration of  
the absolute unit of electromotive      
force1 has been used by ,lorenz to give 
a method of determining resistance which
 is now recognised as the best of all4  
,it is sketched here that the reader may
 obtain some idea of later work on this 
very important subject2 work which is a 
continuation of that of the original    
,british ,association ,committee by     
their successors4 ,a circuit is made up 
of a standard coil of wire1 the ends of 
which are made to touch at the          
circumference and near the centre of the
 disk1 which is placed symmetrically    
with respect to a cylindrical coil1 and 
within it4 ,a current is sent round this
 coil from a battery1 and produces a    

magnetic field within the coil1 the     
lines of magnetic force of which pass   
across the plane of the disk4 ,this     
current1 or a measured fraction of it1  
is also made to flow through the        
standard coil4 ,the disk is now turned  
at a measured speed about its axis1 so  
that the electromotive force due to the 
cutting of the field tends to produce a 
current in the standard coil of wire4   
,the electromotive force of the disk is 
made to oppose the potential difference 
between the ends of this coil due to the
 current1 so that no current flows along
 the disk or the wires connecting it    
with the standard coil4 ,the magnetic   
field within the coil can be calculated 
from the form and dimensions of the coil
 and the current in it 7supposed for the
 moment to be known71 and the           
electromotive force of the disk is      
obtained in terms of its dimensions and 
its speed and the field intensity4 ,but 
this electromotive force1 which is      
proportional to the current in the coil1

 is equal to the product of the         
resistance of the wire and the same     
current1 or a known fraction of it4     
,thus the current appears on both sides 
of the equation and goes out1 and the   
value of the resistance is found in     
absolute units4  ,lord ,rayleigh        
obtained1 by this method1 a result which
 showed that the ,b4,a4 unit was #a4cbc 
per cent4 too small2 and exact          
experiments have been made by others    
with concordant results4 ,values of the 
units have been agreed on by            
,international ,congresses as exact     
enough for general work1 and with these 
units all electrical researches1        
wherever made1 are available for use by 
other experimenters4  ,a vast amount of 
work has been done on this subject      
during the last forty years1 and though 
the value of the practical unit of      
resistance--#aji ,c4,g4,s4 units1 now   
called the ohm--is taken as settled1 and
 copies can now be had in resistance    
boxes1 or separately1 adjusted with all 

needful accuracy1 at the ,national      
,physical ,laboratory and at the ,bureau
 of ,standards at ,washington1 and      
elsewhere1 experiments are being made on
 the exact measurement of currents2     
while a careful watch is kept on the    
standards laid up at these places to see
 whether any perceptible variation of   
their resistance takes place with lapse 
of time4  ,the ,british ,association    
,committee also worked out a complete   
system of units for all electrical and  
magnetic quantities1 and gave the first 
systematic statement of their relations1
 that is1 of the so-called dimensional  
equations of the quantities4 ,this will 
be found in the works to which reference
 has already been made 7p4 #bea74       
,,chapter ,,xiv  ,,the ,,baltimore      
,,lectures   ,the ,baltimore ,lectures  
were delivered in #ahhd at ,johns       
,hopkins ,university1 soon after the    
,montreal meeting of the ,british       
,association4 ,the subject chosen was   
the ,wave ,theory of ,light2 and the    

idea underlying the course was to       
discuss the difficulties of this theory 
to ,professorial fellow-students in     
physical science4 ,a stenographic report
 of the course was taken by ,mr4 ,a4 ,s4
 ,hathaway1 and was published soon      
after4 ,the lectures were revised by    
,lord ,kelvin1 and the book now known as
 ,the ,baltimore ,lectures was published
 just twenty years later 7in #aijd7 at  
the ,cambridge ,university ,press4 ,it  
is absolutely impossible in such a      
memoir as the present to give any       
account of the discussions contained in 
the lectures as now published4 ,the     
difficulties dealt with can for the most
 part only be understood by those who   
are acquainted with the wave theory of  
light in its details1 and such readers  
will naturally go direct to the book    
itself4  ,some of the difficulties1     
however1 were frequently alluded to in  
,lord ,kelvin's ordinary lectures1 and  
all his old students will remember the  
animation with which he discussed the   

apparent anomaly of a medium like the   
luminiferous ether1 which is of such    
enormous rigidity that 7on the elastic  
solid theory7 a wave of transverse      
oscillation is propagated through it    
with a speed of #c  #ajaj centimetres   
7#ahf1jjj miles7 per second1 and yet    
appears to offer no impediment to the   
slow motion of the heavenly bodies4 ,for
 ,lord ,kelvin adopted the elastic solid
 theory of propagation of light as the  
only tenable foundation for the wave    
theory of light in the present state of 
our knowledge1 and dismissed the        
electromagnetic theory 7his words were  
spoken in #ahhd1 it is to be remembered7
 with the statement of his strong view  
that an electric displacement           
perpendicular to the line of            
propagation1 accompanied by a magnetic  
disturbance at right angles to both1 is 
inadmissible4  ,and he goes on to say   
that when we have an electromagnetic    
theory of light1 electric displacement  
will be seen as in the direction of     

propagation1 with ,fresnelian vibrations
 perpendicular to that direction4 ,in   
the preface1 of date ,january #aijd1 the
 insufficiency of the elastic solid     
theory is admitted1 and the question of 
the electromagnetic theory again        
referred to4 ,he says there that the    
object of the ,baltimore ,lectures was  
to ascertain how far the phenomena of   
light could be explained within the     
limits of the elastic solid theory4 ,and
 the answer is everything non-magnetic2 
nothing magnetic4 ,but he adds1 ,the    
so-called electromagnetic theory of     
light has not helped us hitherto1 and   
that the problem is now fully before    
physicists of constructing a            
comprehensive dynamics of ether1        
electricity1 and ponderable matter which
 shall include electrostatic force1     
magnetostatic force1 electromagnetism1  
electrochemistry1 and the wave theory of
 light4  ,all this is exceedingly       
interesting1 for it seems to make clear 
,lord ,kelvin's attitude with respect to

 the electromagnetic theory of ,maxwell1
 which is now regarded by most          
physicists as affording on the whole a  
satisfactory account1 if not a dynamical
 theory in the sense understood by ,lord
 ,kelvin1 of light-propagation4 ,that   
there is an electric displacement       
perpendicular to the direction of       
propagation and a magnetic displacement 
7or motion7 perpendicular to both seems 
proved by the experiments of ,hertz1 and
 the velocity of propagation of these   
disturbances has been found to be that  
of light4 ,of course it remains to be   
found out in what the electric and      
magnetic changes consist1 and whether   
the ether has or has not an atomic      
structure4 ,towards the answer to this  
question on electromagnetic             
presuppositions some progress has       
already been made1 principally by       
,larmor4 ,and1 after all1 while we may  
imagine that we know something more     
definite of dynamical actions on        
ponderable matter1 it is not quite      

certain that we do3 we are more familiar
 with them1 that is almost all4 ,we     
know1 for example1 that at every point  
in the gravitational field of the earth 
we may set up a gravitation vector1 or  
field-intensity2 for a particle of      
matter there is subjected to            
acceleration along that direction4 ,but 
of the rationale of the action we know  
nothing1 or next to nothing4 ,so we set 
up electric and magnetic vectors in an  
insulating medium1 corresponding to     
electric and magnetic effects which we  
can observe2 and it is not too much to  
say that we know hardly less in this    
case than we do in the other1 of the    
inner mechanism of the action of which  
we see the effects4  ,returning to the  
difficulty of the elastic solid theory1 
that while its rigidity is enormous1 it 
offers no obstacle to the planets and   
other heavenly bodies which move through
 it1 it may be interesting to recall how
 ,lord ,kelvin used to deal with it in  
his elementary lectures4 ,the same      

discussion was given in the             
,introductory ,lecture at ,baltimore4   
,the difficulty is not got over by an   
explanation of what takes place3 it is  
turned by showing that a similar        
difficulty exists in reconciling        
phenomena which can be observed every   
day with such ordinary materials as     
pitch or shoemakers' wax4 ,a piece of   
such wax can be moulded into a          
tuning-fork or a bell1 and will then1 if
 struck1 sound a musical note of        
definite pitch4 ,this indicates1 for    
rapidly alternating deformations started
 by a force of short duration1 the      
existence of internal forces of the kind
 called elastic1 that is1 depending on  
the amount of deformation caused1 not on
 the rate at which the deformation is   
increasing or diminishing1 as is the    
case for the so-called viscous forces   
which are usually displayed by such     
material4 ,but the tuning-fork or bell1 
if left lying on the table1 will        
gradually flatten down into a thin sheet

 under only its own weight4 ,here the   
deformation is opposed only by viscous  
forces1 which1 as the change is very    
slow1 are exceedingly small4  ,but let a
 large slab of it1 three or four inches 
thick1 be placed in a glass jar ten or  
twelve inches in diameter1 already      
partly filled with water1 and let some  
ordinary corks be imprisoned beneath1   
while some lead bullets are laid on the 
upper surface4 ,after a month or two it 
will be found that the corks have       
disappeared from the water into the wax1
 and that the orifices which they made  
in entering it have healed up           
completely2 similarly the bullets have  
sunk down into the slab1 leaving no     
trace behind4 ,after two or three months
 more1 the corks will be seen to be     
bursting their way out through the upper
 surface of the slab1 and the bullets   
will be found in the water below4 ,the  
very thing has taken place that would   
have happened if water had been used    
instead of pitch1 only it has taken a   

very much longer time to bring it about4
 ,the corks have floated up through the 
wax in consequence of hydrostatic upward
 force exerted by the wax acting as a   
fluid2 and the bullets have sunk down in
 consequence of the excess of their     
weights above the upward hydrostatic    
force exerted on them as on the corks4  
,the motion in both cases has been      
opposed by the viscous forces called    
into play4  ,the application of this to 
the luminiferous ether is immediate4    
,let the ether be regarded as a         
substance which can perform vibrations  
only when times and forces are suitable1
 that is1 when the forces producing     
distortion act for only an infinitesimal
 time 7as in the starting of the        
tuning-fork by a small blow71 and are   
not too great4 ,vibrations may be set up
 locally1 and the medium may have a true
 rigidity by which they are propagated  
to more remote parts2 that is to say1   
waves travel out from the centre of     
disturbance4 ,on the other hand1 if the 

forces are long continued1 even if they 
be small1 they produce continuously     
increasing change of shape4 ,thus the   
planets move seemingly without          
resistance4  ,the conclusion is that the
 apparently contradictory properties of 
the ether are no more mysterious than   
the properties of pitch or shoemakers'  
wax4 ,and1 after all1 matter is still a 
profound mystery4  ,dynamical           
illustrations1 which old ,glasgow       
students will recognise1 appear         
continually in the lectures4 ,they will 
remember1 almost with affection1 the    
system of three particles 7#g lb4 or #ad
 lb4 weights67 joined together in a     
vertical row by stout spiral springs of 
steel1 which were always to be taken as 
massless1 and will recall ,lord         
,kelvin's experiments with them1        
demonstrating the three modes of        
vibration of a system of three masses1  
each of which influenced those next it  
on the two sides4 ,here they will find  
the problem solved for any number of    

particles and intervening springs1 and  
the solution applied to an extension of 
the massive molecule which von          
,helmholtz imbedded in the elastic      
ether1 and used to explain anomalous    
dispersion4 ,a highly complex molecule  
is suggested1 consisting of an outer    
shell embedded in the ether as in the   
simpler case1 a second shell within that
 connected to the outer by a sufficient 
number of equal radial springs1 a third 
within and similarly connected to the   
second by radial springs1 and so on4    
,this molecule will have as many modes  
of vibration as there are sets of       
springs1 and can therefore impart1 if it
 is set into motion1 a complex          
disturbance to the ether in which it is 
imbedded4  ,the modification of this    
arrangement by which ,lord ,kelvin      
explained the phosphorescence of such   
substances as luminous paint is also    
described1 and will be recognised by    
some as an old friend4 ,a number1 two   
dozen or so1 of straight rods of wood   

eighteen inches long are attached to a  
steel wire four or five inches apart1   
like steps on a ladder made with a      
single rope along the centres of the    
steps4 ,the wire is so attached to each 
rod that the rod must turn with the wire
 if the latter is twisted round4 ,each  
rod is loaded with a piece of lead at   
each end to give it more moment of      
inertia about the wire4 ,the wire1 with 
this ladder attached to it1 is rigidly  
attached to the centre of a cross-bar at
 the top1 which can be made to swing    
about the wire as an axis and so impart 
twisting vibrations to the wire in a    
period depending on this driver4        
,sliding weights attached to the bar    
enable its moment of inertia to be      
changed at pleasure4 ,the lower end of  
the wire carries a cross-bar with two   
vanes1 immersed in treacle in a vessel  
below4 ,when the period of the exciter  
was very long the waves of torsion did  
not travel down the ladder1 but when the
 period was made sufficiently short the 

waves travelled down and were absorbed  
in the treacle below4 ,in the former    
case the vibrations persisted2 the case 
was analogous to that of                
phosphorescence4  ,illustration3 ,,fig4 
#ah4  ,incidentally a full and very     
attractive account of the elastic solid 
theory is given in these lectures1      
accompanied as it is by characteristic  
digressions on points of interest which 
suggest themselves1 and on topics on    
which the lecturer held strong opinions1
 such1 for example1 as the absurd       
,british system of weights and measures4
 ,the book reads in many places like a  
report of some of the higher            
mathematical lectures which were given  
every session at ,glasgow2 and on that  
account1 if on no other1 it will be read
 by the old students of the higher class
 with affectionate interest4 ,but the   
discussions of the great fundamental    
difficulty presented at once by         
dispersion--the fact1 that is1 that     
light of different wave lengths has     

different velocities in ordinary        
transparent matter--the discussions of  
the various theories of dispersion that 
have been put forward1 the construction 
of the molecules1 gyrostatic and        
non-gyrostatic1 with all their          
remarkable properties1 which ,lord      
,kelvin invents in order to frame a     
dynamical mechanism which will imitate  
the action of matter as displayed in the
 complex manifestations of the optical  
phenomena1 not only of isotropic matter1
 but of crystals1 will ever afford      
instruction to every mathematician who  
has the courage to attack this subject1 
and remain as a monument to the         
extraordinary genius of their author4   
,a subject is touched on in these       
lectures which has not been dealt with  
in the present review of ,lord ,kelvin's
 work4 ,by four lines of argument--by   
the heat of combination of copper and   
zinc1 together with the difference of   
electric potential developed when these 
metals are put in contact1 from the     

thickness of a capillary film of soap   
and water 7measured by ,rcker and       
,reinold7 just before it gives way1 and 
the work spent in stretching it1 from   
the kinetic theory of gases and the     
estimated length of free path of a      
particle 7given also by ,loschmidt and  
by ,johnstone ,stoney71 and from the    
undulatory theory of light--,lord       
,kelvin estimated superior and inferior 
limits to the size of the atoms of      
bodies1 or1 more properly speaking1 of  
the molecular structure of the matter4  
,we cannot discuss these arguments--and 
they can be read at leisure by any one  
who will consult ,volume ,i             
7,constitution of ,matter7 of ,lord     
,kelvin's ,popular ,lectures and        
,addresses1 for his ,royal ,institution 
,lecture on the subject1 there given in 
full--but we may state his conclusion4  
,let a drop of water1 a rain drop1 for  
example1 be magnified to the size of the
 earth1 that is1 from a sphere a quarter
 of an inch1 or less1 in diameter to a  

sphere #hjjj miles in diameter1 and let 
the dimensions of the molecular         
structure be magnified in the same      
proportion4 ,the magnified structure    
would be more coarse-grained than a heap
 of small shot1 but probably less       
coarse-grained than a heap of           
cricket-balls4  ,of course1 it is not   
intended here to convey the idea that   
the molecules are spheres like shot or  
cricket-balls2 they undoubtedly have a  
structure of their own4 ,and no         
pronouncement is made as to the         
divisibility or non-divisibility of the 
molecules4 ,all that is alleged is that 
if the division be carried to a         
minuteness near to or beyond that of the
 dimensions of the structure1 portions  
of the substance will be obtained which 
have not the physical properties of the 
substance in bulk4  ,the recent         
interesting researches of chemists and  
physicists into phenomena which seem to 
demonstrate the disintegration1 not     
merely of molecules1 but even of the    

atomic structure of matter1 attracted   
,lord ,kelvin's attention in his last   
years1 and suo more he endeavoured to   
frame dynamical explanations of         
electronic 7or1 as he preferred to call 
it1 electrionic7 action4 ,but though    
keenly interested in all kinds of       
research1 he turned again and again to  
the older theories of light1 and his    
dynamical representations of the ether  
and of crystals1 with renewed vigour and
 enthusiasm4     ,,chapter ,,xv  ,,speed
 ,,of ,,telegraph ,,signalling--,,laying
 ,,of ,,submarine ,,cables--,,telegraph 
,,instruments--,,navigational           
,,instruments1 ,,compass ,,and          
,,sounding ,,machine   ,,theory ,,of    
,,signalling  ,when the question of     
laying an ,atlantic cable began to be   
debated in the middle of the nineteenth 
century1 ,professor ,thomson undertook  
the discussion of the theory of         
signalling through such a cable4 ,it was
 not generally understood by practical  
telegraphists that the conditions of    

working would be very different from    
those to which they were accustomed on  
land lines1 and that the instruments    
employed on such lines would be useless 
for a cable4 ,such a cable consists of a
 copper conductor separated from the    
sea-water by a coating of gutta-percha2 
it forms an elongated ,leyden jar of    
very great capacity1 which1 when a      
battery is connected to one end of the  
conducting core1 is gradually charged   
up1 first at that end1 and later and    
later at greater distances from it1 and 
then is gradually discharged again when 
the battery is withdrawn and the end of 
the conductor connected to earth4 ,here1
 again1 an application of ,fourier's    
analysis solved the problem1 which1 with
 certain modifications1 and on the      
supposition that the working is slow1 is
 essentially the same problem as the    
diffusion of heat along a conducting    
bar1 or the diffusion of a salt solution
 along a column of water4 ,the signals  
are retarded 7and this was one of the   

results of the investigation7 in such a 
manner that the time required to reach a
 stated fraction of the maximum strength
 of current at the remote end1 when a   
given potential difference is applied at
 the other1 or home end1 is proportional
 to the product of the capacity and     
resistance of the cable1 each taken per 
unit of the length1 and also            
proportional to the square of the length
 of cable4 ,in other words1 the         
retardation is proportional to the      
product of the resistance of the copper 
conductor and the total capacity of the 
cable4 ,this gave a practical rule of   
great importance for guidance in the    
manufacture of submarine cables4 ,the   
conductor should have the highest       
conductivity obtainable1 and should     
therefore be of pure copper2 the        
insulating covering should1 while       
forming a nearly absolutely             
non-conducting sheath1 have as low a    
specific inductive capacity as possible4
 ,the first of these conditions ran     

counter to some views that had been put 
forward1 to the effect that it was only 
necessary to have the internal conductor
 highly conducting on its surface2 and  
some controversy on the subject ensued4 
,the inverse square law1 as it was      
called1 was vehemently called in        
question1 from a mistaken interpretation
 of some experiments that were made to  
test it4 ,for if the potential at the   
home end be regularly altered1 according
 to the simple harmonic law1 so that the
 number of periods of oscillation in a  
second is n1 the changes of potential   
are propagated with velocity #b7ncr71   
where c and r are the capacity and      
resistance of the cable1 each taken per 
unit length4 ,in this case1 for a long  
cable1 there is a velocity of           
propagation independent of the length2  
and this fact seems to have misled the  
experimenters4 ,thomson's view          
prevailed1 and the result was the       
establishment1 first by ,thomas ,bolton 
,sons1 ,stoke-on-,trent1 of mills for   

the manufacture of high conductivity    
copper1 which is now a great industry4  
,the ,fourier mathematics of the        
conduction of heat along a bar suffices 
to solve the problem1 so long as the    
signalling is so slow as not to bring   
into play electromagnetic induction to  
any serious extent4 ,for rapid          
signalling in which very quick changes  
of current are concerned the            
electromotive forces due to the growth  
or dying out of the current would be    
serious1 and the theory of diffusion    
would not apply4 ,but ordinary cable    
working is quite slow enough to enable  
such electromotive forces to be         
disregarded4   ,,laying ,,of ,,first    
,,american ,,cables  ,the first cable of
 #aheh was laid by the ,u4,s4 frigate   
,niagara and ,h4,m4,s4 ,agamemnon1 after
 having been manufactured with all the  
precautions suggested by ,professor     
,thomson's researches4 ,it is hard to   
realise how difficult such an enterprise
 was at the time4 ,the manufacture of a 

huge cable1 the stowage of it in cable  
tanks on board the vessels1 the         
invention of laying and controlling and 
picking-up machinery had to be faced    
with but little experience to guide the 
engineers4 ,here again ,thomson1 by his 
knowledge of dynamics and true          
engineering instinct1 was of great      
assistance4 ,in #ahfe he read a very    
valuable paper on the forces concerned  
in the laying and lifting of deep-sea   
cables1 showing how the strains could be
 minimised in various practical cases of
 importance--for example1 in the lifting
 of a cable for repairs4  ,a first      
,atlantic cable had been partly laid in 
#aheg by the ,niagara1 when it broke in 
#bjjj fathoms of water1 about #ccj miles
 from ,valentia1 where the laying had   
begun4 ,an additional length of #ijj    
miles was made1 and the enterprise was  
resumed4 ,this time it was decided that 
the two vessels1 each with half of the  
cable on board1 should meet and splice  
the cable in mid-ocean1 and then steam  

in opposite directions1 the ,agamemnon  
towards ,valentia1 the ,niagara towards 
,newfoundland4 ,professor ,thomson was  
engineer in charge of the electrical    
testing on board of the ,agamemnon4     
,after various mishaps the cable was at 
last safely laid on ,august #f1 #aheh1  
and congratulations were shortly after  
exchanged between ,great ,britain and   
the ,united ,states4 ,on ,september #f  
it was announced that signals had ceased
 to pass1 and an investigation of the   
cause of the stoppage was undertaken by 
,professor ,thomson and the other       
engineers4 ,the report stated that the  
cable had been too hastily made1 that1  
in fact1 it was not good enough1 and    
that the strains in laying it had been  
too great and unequal4 ,it was found    
impossible to repair it1 so that there  
was no option but to abandon it4  ,this 
cable probably suffered seriously from  
the violent means which seem to have    
been employed to force signals through  
it4 ,now only a very moderate difference

 of potential is applied to a cable at  
the sending end1 and speed of signalling
 is obtained by the use of instruments1 
the moving parts of which have little   
inertia1 and readily respond to only an 
exceedingly feeble current4  ,a second  
cable was made and laid in #ahfe by the 
,great ,eastern1 which could take on    
board the whole at once and steam from  
shore to shore4 ,it was also well       
adapted for cable work through having   
both screw and paddles4 ,as ,thomson    
points out1 steerage way could be got on
 the vessel by driving the screw ahead1 
so as to send a stream of water astern  
towards the rudder1 while the paddles   
were driven astern to prevent the ship  
from going ahead4 ,this was of great    
advantage in manuvring on many          
occasions4  ,this cable also broke1 but 
a third was laid successfully in #ahff  
by the same vessel1 and the second was  
recovered and repaired1 so that two good
 cables were secured for commercial     
working4 ,on both expeditions ,professor

 ,thomson acted as electrical engineer1 
and received the honour of knighthood   
and the thanks of the ,anglo-,american  
,telegraph ,company on his return home1 
when he was also presented with the     
freedom of the city of ,glasgow4  ,he   
afterwards acted as engineer for the    
,french ,atlantic ,cable1 for the       
,brazilian and ,river ,plate ,company1  
and for the ,commercial ,company1 whose 
two new ,atlantic cables were laid in   
#ahhb-d4   ,,mirror ,,galvanometer ,,and
 ,,siphon ,,recorder  ,since whatever   
the potential applied at the sending end
 of the cable might be 7and1 of course1 
as has been stated1 this potential had  
to be kept to as low a value as         
possible7 the current at the receiving  
end only rose gradually1 it was         
necessary to have as delicate a         
receiving instrument as possible1 so    
that it would quickly respond to the    
growing and still feeble current4 ,for  
unless the cable could be worked at a   
rate which would permit of charges per  

word transmitted which were within the  
reach of commercial people1 it was      
obvious that the enterprise would fail  
of its object4 ,and as a cable could not
 cost less than half a million sterling1
 the revenue to be aimed at was very    
considerable4 ,this problem ,thomson    
also solved by the invention of his     
mirror galvanometer4 ,the suspended     
magnet was made of small pieces of      
watch-spring cemented to a small mirror1
 so that the whole moving part weighed  
only a grain or two4 ,its inertia1 or   
resistance to being set into motion1 was
 thus very small1 and it was hung by a  
single fibre of silk within a closed    
chamber at the centre of the            
galvanometer coil4 ,a ray of light from 
a lamp was reflected to a white paper   
scale in front of the mirror1 which as  
it turned caused a spot of illumination 
to move along the paper4 ,a motion of   
this long massless index to the left was
 regarded as a dot1 a motion to the     
right as a dash1 and the ,morse alphabet

 could therefore be employed4 ,this     
instrument was used in the #aheh cable  
expedition1 and a special form of       
suspension was invented for it by       
,thomson1 to enable it to be used on    
board ship4 ,the suspension thread1     
instead of being held at one end only1  
was stretched from top to bottom of the 
chamber in which the needle hung1 and   
kept tight by being secured at both     
ends4 ,thus the minimum of disturbance  
was caused to the mirror by the rolling 
or pitching of the ship4  ,the          
galvanometer was also enclosed in a     
thick iron case to guard it against the 
magnetic field due to the iron of the   
ship4 ,the iron-clad galvanometer first 
used in submarine telegraphy 7on the    
#aheh expedition in the ,u4,s4 frigate  
,niagara7 is in the collection of       
historical apparatus in the ,natural    
,philosophy ,department of the          
,university of ,glasgow4  ,the mirror   
galvanometer then invented has become   
one of the most useful instruments of   

the laboratory4 ,mirror deflection is   
now used also for the indicators of many
 kinds of instruments4  ,the            
galvanometer was replaced later by      
another invention of ,professor         
,thomson--the siphon recorder4 ,here a  
small and delicate pen was formed by a  
piece of very fine glass tube           
7vaccination tubing1 in fact7 in the    
form of a siphon1 of which the shorter  
end dipped into an ink-bottle1 while the
 other end wrote the message in little  
zig-zag notches on a ribbon of paper    
drawn past it by machinery4 ,the siphon 
was moved to and fro by the signalling  
currents1 which flowed in a small coil  
hung between the poles of an            
electromagnet1 excited by a local       
battery1 and the ink was spirted in a   
succession of fine drops from the pen to
 the paper4 ,this was accomplished by   
electrifying the ink-bottle and ink by a
 local electrical machine1 and keeping  
the paper in contact with an uninsulated
 metal roller4 ,electric attraction     

between the electrified ink and the     
unelectrified paper thus drew the       
ink-drops out1 and the pen1 which never 
touched the paper1 was quite unretarded 
by friction4 ,both these instruments had
 the inestimable advantage that the to  
and fro motions of the spot of light or 
the pen took place independently of     
ordinary earth-currents through the     
cable4  ,the arrangement of magnet and  
suspended coil in this instrument has   
become widely known as that of the      
d',arsonval galvanometer4 ,this         
application was anticipated by ,thomson1
 and is distinctly mentioned in his     
recorder patent1 long before such       
galvanometers were ever used4 ,it was   
later proposed by several experimenters 
before ,m4 d',arsonval4  ,it is not too 
much to say that1 by his discussion of  
the speed of signalling1 his services as
 an electrical engineer1 and especially 
by his invention of instruments capable 
of responding to very feeble currents1  
,thomson made submarine telegraphy      

commercially possible4 ,later he entered
 into partnership with ,mr4 ,c4 ,f4     
,varley and ,professor ,fleeming        
,jenkin4 ,a combination of inventions   
was made by the firm3 ,varley had       
patented a method of signalling by      
condensers1 and ,jenkin later suggested 
and patented an automatic key for       
curb-sending on a cable--that is1       
signalling by placing one pole of the   
battery for an interval a little shorter
 than the usual one to the line1 and    
then reversing the battery for the      
remainder4 ,this gave sharper signals1  
as the reversal helped to discharge the 
cable more rapidly than it would have   
been by the mere connection to earth    
between two signals4 ,the firm of       
,thomson1 ,varley  ,jenkin took a       
prominent part in cable work2 and       
,thomson and ,jenkin acted as engineers 
for many large undertakings4 ,they      
employed a staff of young electricians  
at the cable-works at ,millwall and     
elsewhere1 keeping watch over the cable 

during manufacture1 and sent them to sea
 as representatives and assistants to   
perform similar duties during the       
process of cable-laying4 ,on their staff
 were many men who have come to eminence
 in electrical and engineering pursuits 
in later life4   ,,mariners' ,,compass  
,,and ,,sounding ,,machine  ,after the  
earlier ,atlantic expeditions ,sir      
,william ,thomson turned his attention  
to the construction of navigational     
instruments1 and invented the mariner's 
compass and wire-sounding apparatus     
which are now so well known4 ,he had    
come to the conclusion that the         
compasses in use had much too large     
needles 7some of them bar-magnets seven 
or eight inches long67 to respond       
quickly and certainly to changes of     
course1 and1 what was still more        
serious1 to admit of the application of 
correcting magnets1 and of masses of    
soft-iron to annul the action of the    
magnetism of the ship4  ,the compass    
card consists of a paper ring1 on which 

the points and degrees are engraved in  
the ordinary way1 and is kept circular  
by a light ring of aluminium4 ,threads  
of silk extend radially from the rim to 
a central boss of aluminium in which is 
a cap of aluminium4 ,in the top of the  
cap is a sapphire bearing1 which rests  
on an iridium point projecting upward   
from the compass bowl4 ,eight magnets of
 glass-hard steel1 from #c inches to #b 
inches long1 and about the thickness of 
a knitting-needle1 which form the       
compass needle1 are strung like the     
steps of a rope ladder1 on two silk     
threads attached to four of the radial  
threads4  ,the weight of the card is    
extremely small--only #agj grains2 that 
is less than  of an ounce4 ,but the     
matter is not merely made small in      
amount2 it is distributed on the whole  
at a great distance from the axis2      
consequently the period of free         
vibration is long1 and the card is very 
steady4 ,the great lightness of the card
 also causes the error due to friction  

on the point of support to be very      
small4  ,the errors of the compass in an
 iron ship are mainly the semicircular  
error and the quadrantal error4 ,we can 
only briefly indicate how these arise   
and how they are corrected4 ,the ship's 
magnetism may be considered as partly   
permanent1 and partly inductive4 ,the   
former changes only very slowly1 the    
latter alters as the ship changes course
 and position4 ,for the ship is a       
combination of longitudinal1 transverse1
 and vertical girders and beams4 ,as a  
whole it is a great iron or steel       
girder1 but its structure gives it      
longitudinal1 transverse1 and vertical  
magnetisation4 ,this disturbs the       
compass1 which is also affected by the  
magnetisation of the iron or steel masts
 and spars1 or of iron or steel carried 
as cargo4  ,the semicircular error is   
due to a great extent to permanent      
magnetism1 but also in part to induced  
magnetism4 ,it is so called because when
 the ship's head is turned through #cfj1

 the error attains a maximum on two     
courses #ahj apart4 ,it may amount to   
over #bj in an ordinary iron vessel1 and
 to #cj or #dj in an armour-clad4 ,it is
 corrected by two sets of steel magnets 
placed with their centres under the     
needle in the binnacle4 ,one set have   
their lengths fore and aft1 the others  
in the thwart-ship direction4 ,these    
magnets annul the error on the north and
 south and on the east and west courses1
 due to the two horizontal components of
 magnetic force produced mainly by the  
permanent magnetism of the ship4 ,a     
regular routine of swinging the ship    
when marks on the shore 7the true       
bearings of which from the ship are     
known7 are available1 is followed for   
the adjustment4  ,the quadrantal error  
is so called because its maxima are     
found on four compass courses           
successively a quadrant1 or #ij1 from   
one another4 ,it amounts in general to  
from #e to #aj at most4 ,it is due to   
induced magnetism1 and is corrected by a

 pair of soft-iron spheres1 placed on   
the two sides of the compass with their 
centres in a line transverse to the     
ship1 through the centre of the compass 
needle4 ,there are1 however1 exceptional
 cases in which they are placed in the  
fore and aft line one afore1 the other  
abaft1 the needle4 ,when the quadrantal 
error has once been annulled it is      
always zero1 for as the induced         
magnetism changes1 so does that of the  
spheres1 and the adjustment remains     
good4 ,in a new ship the permanent      
magnetism slowly alters1 and so the     
semicircular correction has to be       
improved from time to time by changing  
the magnets4  ,these adjustments are not
 quite all that have to be made2 but    
enough has been stated to show how the  
process of compensation can be carried  
out with the ,thomson compass4 ,the     
immensely-too-large magnets used        
formerly as compass needles1 through a  
mistaken notion1 apparently1 that more  
directive force would be got by their   

means1 rendered the quadrantal          
adjustment an impossibility4 ,the card  
swinging round brought the large needles
 into different positions relatively to 
the iron balls1 when these were used1   
and exerted an inductive action on them 
which reacted on the needles1 producing 
more error1 perhaps1 than was corrected4
  ,thomson invented also an instrument  
called a deflector1 by which it is      
possible to adjust a compass when sights
 of sun or stars1 or bearings of        
terrestrial objects1 cannot be obtained4
 ,by means of it the directive forces on
 the needles on different courses can be
 compared4 ,then the adjustment is made 
by placing the correctors so that the   
directive force is as nearly as may be  
the same on all courses4 ,the compass is
 then quite correct4  ,the theory of    
deviations of the compass1 it is right  
to say1 was discussed first partially by
 ,poisson1 but afterwards very          
completely and elegantly by the late    
,mr4 ,archibald ,smith of ,jordanhill1  

whose memoirs1 now incorporated in the  
,admiralty ,manual of ,deviations of the
 ,compass1 led to ,lord ,kelvin's       
inventions4  ,lord ,kelvin's compass is 
now almost universally in use in the    
merchant service of this country1 and in
 most of the navies of the world4 ,it   
has added greatly to the certainty and  
safety of navigation4  ,the sounding    
machine is also well known4 ,at first   
pianoforte wire was used for deep-sea   
sounding by ,commodore ,belknap of the  
,u4,s4 ,navy1 and by others1 on ,sir    
,william ,thomson's recommendation4     
,finally1 a form of machine was made by 
which a sinker could be lowered to the  
bottom of the sea and brought up again  
in a few minutes2 so that it was        
possible to take a sounding without the 
long delay involved in the old method   
with a reel of hemp-rope1 which often   
tempted shipmasters to run risks of     
going ashore rather than stop the ship  
for the purpose4 ,the wire offered      
little resistance to motion through the 

water1 and by a proper winding machine1 
with brake to prevent the wire from     
running out too fast and kinking1 when  
it was almost certain to break1 one man 
could quickly sound and heave up again1 
while another attended to the wire and  
sinker4 ,a gauge consisting of a long   
quill-tube closed at the upper end1 and 
coated inside with chromate of silver1  
showed by the action of the sea-water on
 the coating how far the water had      
passed up the tube1 compressing the air 
above it2 and from this1 by placing the 
tube along a wooden rule properly       
graduated1 the depth was read off at    
once4 ,with the improved machine a ship 
approaching the shore in thick weather  
could take soundings at short intervals 
without stopping1 and discover at once  
any beginning of shallowing of the      
water1 and so avoid danger4  ,the single
 wire is not now used1 as a thin        
stranded wire is found safer and quite  
as effective4 ,the gauge also has been  
improved4 ,the apparatus can be seen in 

any well-found sea-going vessel2 though 
there are still1 or were until not very 
long ago1 steam vessels without this    
apparatus1 though crossing the ,english 
,channel with passengers4 ,these        
depended for soundings on the obsolete  
hemp-rope1 wrapped round an iron spindle
 held vertically on the deck by members 
of the ship's company1 while the cord   
was unwound by the descent of the       
sinker4#bd  ,sir ,william ,thomson's    
electrical and other inventions are too 
numerous to specify here1 and they are  
in constant use wherever precision of   
measurement is aimed at or required4    
,long ago he invented electrometers for 
absolute measurements of electrical     
potential 7electric pressure72 more     
recently his current-balances have given
 the same precision to electrodynamic   
measurement of currents4 ,all his early 
instruments were made by ,mr4 ,james    
,white1 ,glasgow4 ,the business founded 
by ,mr4 ,white1 and latterly carried on 
at ,cambridge ,street1 has developed    

immensely1 and is now owned by a limited
 liability company--,messrs4 ,kelvin and
 ,james ,white 7,limited74  ,for many   
years ,sir ,william ,thomson was a keen 
yachtsman1 and his schooner yacht1 the  
,lalla ,rookh1 was well known on the    
,clyde and in the ,solent4 ,an expert   
navigator1 he delighted to take deep-sea
 voyages in his yacht1 and went more    
than once as far as ,madeira4 ,many     
navigational and hydrodynamical problems
 were worked out on these expeditions4  
,for a good many years1 however1 he had 
given up sea-faring during his times of 
relaxation1 and lived in ,glasgow and   
,london and in ,largs1 ,ayrshire1 where 
he built1 in #ahge1 a large and         
comfortable house1 looking out towards  
the ,firth and the ,argyleshire lochs he
 knew and loved so well4  ,in the course
 of his deep-sea expeditions in his     
yacht he became impressed with the      
utility of ,sumner's method of          
determining the position of a ship4 ,let
 us suppose that at a given instant the 

altitude of the sun is determined from  
the ship4 ,the ,greenwich meantime1 and 
therefore the longitude at which the sun
 is vertical1 is known by chronometer1  
and the declination of the sun is known 
from the ,nautical ,almanac4 ,the point 
on the earth vertically under the sun   
can be marked on the chart1 and a circle
 7or rather1 what would be a circle on a
 terrestrial globe7 drawn round it from 
every point of which the sun would have 
the observed altitude4 ,the ship is at a
 point on this circle4 ,some time after 
the altitude of the sun is observed     
again1 and a new circle is drawn4 ,if   
the first circle be bodily shifted on   
the chart along the distance run in the 
interval1 it will intersect the second  
in two points1 one of which will be the 
position of the ship1 and it is         
generally possible to tell which1       
without danger of mistake4  ,sir        
,william ,thomson printed tables for    
facilitating the calculations in the use
 of ,sumner's method1 and continually   

used them in his own voyages4 ,he was   
well versed in seamanship of all kinds1 
and used his experience habitually to   
throw light on abstruse problems of     
dynamics4 ,some of these will be found  
in ,thomson and ,tait2 for instance1 in 
,part ,i1  #cbe1 where a number of      
nautical phenomena are cited in         
illustration of an important principle  
of hydrodynamics4 ,the fifth example    
stated is as follows3 ,in a smooth sea1 
with moderate wind blowing parallel to  
the shore1 a sailing ship heading       
towards the shore1 with not enough of   
sail set1 can only be saved from        
creeping ashore by setting more sail1   
and sailing rapidly towards the shore1  
or the danger that is to be avoided1 so 
as to allow her to be steered away from 
it4 ,the risk of going ashore in        
fulfilment of ,lagrange's equations is a
 frequent incident of 'getting under    
way' while lifting anchor or even after 
slipping from moorings4 ,his seamanship 
was well known to shipmasters1 with whom

 he had much intercourse1 and whose     
intelligence and practical skill he held
 in very high regard4     ,,chapter     
,,xvi  ,,lord ,,kelvin ,,in ,,his       
,,class-,,room ,,and ,,laboratory   ,it 
is impossible to convey to those who    
never studied at ,glasgow any clear     
conception of ,thomson as he appeared to
 students whom he met daily during the  
session4 ,his appearance at meetings of 
the ,british ,association1 and his      
vivacious questionings of the various   
authors of papers1 his absorption in his
 subject and oblivion to the flight of  
time when he read a paper himself1 will 
long be remembered by scientific men3   
but though they suffice to suggest what 
he was like in his own lecture-room1 the
 picture lacks the setting of furniture1
 apparatus1 assistants1 and students1   
which all contributed to the unique     
impression made by his personality on   
his pupils4 ,the lecture-table--with    
long straight front and ends refracted  
inward1 flanked by higher small round   

tables supported on cylindrical         
pillars--laden with instruments2 the    
painted diagrams of the solar spectrum  
and of the paths of coloured rays       
through a prism1 hung round the walls2  
the long wire with the cylindrical      
vibrator attached1 for experiments on   
torsion1 and the triple spiral spring   
vibrator1 which hung at the two ends of 
the long blackboard2 the pendulum thirty
 feet long1 consisting of a steel wire  
and a twelve-pound cannon-ball as bob1  
suspended from the apex of the dome-roof
 above the lecture-table2 the large iron
 wheel in the beautiful oriel window on 
the right of the lecturer1 and the      
collection of optical instruments on the
 table in front of the central window   
spaces1 from which the small iron-framed
 panes--dear to the heart of the        
architect--had been removed2 the clock  
on either side of the room1 one         
motionless1 the other indicating the    
time1 and having attached to it the     
alarm which showed when the angry bell  

outside had ceased to toll2 the ten     
benches of eager and merry students1    
which filled the auditorium2 all these  
combined to form a scene which every    
student fondly recalls1 and which cannot
 be adequately described4 ,a similar    
scene1 with some differences of         
arrangement and having its own          
particular associations1 will occur to  
every student who attended in the ,old  
,college4  ,the writer will never forget
 the lecture-room when he first beheld  
it1 from his place on ,bench ,,viii1 a  
few days after the beginning of session 
#ahgd-e4 ,sir ,william ,thomson1 with   
activity emphasised rather than         
otherwise by his lameness1 came in with 
the students1 passed behind the table1  
and1 putting up his eye-glass1 surveyed 
the apparatus set out4 ,then1 as the    
students poured in1 an increasing       
stream1 the alarm weight was released by
 the bell-ringer1 and fell slowly some  
four or five feet1 from the top of the  
clock to a platform below4 ,by the time 

the weight had descended the students   
were in their places1 and then1 as      
,thomson advanced to the table1 all rose
 to their feet1 and he recited the third
 ,collect from the ,morning ,service of 
the ,church of ,england4 ,it was the    
custom then1 and it is still one better 
honoured in the observance than in the  
breach 7which has become rather common7 
to open all the first and second classes
 of the day with prayer2 and the        
selection of the prayers was left to the
 discretion of the professors4 ,next    
came the roll-call by the assistant2    
each name was called in its ,english1 or
 ,scottish 7for the clans were always   
well represented7 form1 and the answer  
adsum was returned4  ,then the          
,professor began his lecture1 generally 
with the examination of one of the      
students1 who rose in his place when his
 name was called4 ,thomson1 as the      
quotation in ,chapter ,,vi from the     
,bangor ,address shows1 was fond of oral
 examination1 and after the second hour 

had begun to decline as one of regular  
attendance1 habitually devoted ten or   
fifteen minutes to asking questions and 
criticising the answers4 ,the names of  
the students to be questioned were      
selected at random from the class       
register1 or by a kind of lottery1      
carried out by placing a small card for 
each student in a box on the table1 and 
drawing a name whenever a member of the 
class was to be examined4 ,the interest 
in the drawing each day was intense1 for
 there was a glorious uncertainty as to 
what might be the line of examination   
adopted4 ,sometimes1 in the midst of a  
criticism of an answer1 an idea would   
suddenly occur to the ,professor1 and he
 would enlarge upon it1 until the       
forgotten examinee slipped quietly back 
into his seat1 to be no more disturbed  
at least for that day6 ,and how great   
the relief if the ordeal was well passed
 and the card was placed in that        
receptacle of the blessed1 the          
compartment reserved for those who had  

been called and duly passed the assize6 
,but there was a third compartment      
reserved for the cards of those         
unfortunates who failed to satisfy the  
judge6 ,the reader may have anticipated 
the fact that the three divisions of    
this fateful box were commonly known to 
students by the names of the three great
 habitations of spirits described in the
 ,divina ,commedia of ,dante4  ,as has  
been stated1 the oral examination with  
which the lectures opened was the cause 
of a good deal of excitement1 which was 
added to by the element of chance       
introduced by drawing the names from the
 purgatorial compartment of the box4    
,the ordeal was dreaded by backward     
students1 whom ,thomson found1 as he    
said1 aphasic1 when called on to answer 
in examination1 but who certainly were  
anything but aphasic in more congenial  
circumstances4 ,occasionally they       
abstained from responding to their      
names1 modestly seeking the seclusion of
 the crowd1 and some little time would  

be spent in ascertaining whether the    
examinee-designate was present4 ,when at
 last he was discovered1 he generally   
rose with a fervent appeal to his       
fellows on either side to help him in   
his need4  ,mc,farlane used to tell of  
an incident which illustrated the       
ingenuity with which it was sometimes   
attempted to evade the ordeal of the    
viva voce examination4 ,one afternoon1  
when he was busily preparing the        
lecture-illustrations for next day1 a   
student came into the class-room1 and   
engaging him in conversation on some    
point of dynamics1 regarding which he   
professed to have a difficulty1 hovered 
round the box which contained the three 
compartments popularly known as         
,purgatory1 ,heaven1 and ,hell6 ,always 
when ,mc,farlane left the room to bring 
something from the adjoining cabinet of 
apparatus1 he found1 when he returned1  
his inquiring friend hurriedly quitting 
the immediate vicinity of the box4 ,at  
last the student took leave1 with many  

apologies for giving so much trouble4   
,as ,mc,farlane suspected would be the  
case1 the ticket bearing the name of    
that student was no longer to be found6 
,he used to conclude the story as       
follows3 ,i just made a new ticket for  
him1 and placed it on the top of the    
other tickets1 and next day ,sir        
,william called him1 the very first     
time4 ,what were his feelings1 who had  
fondly thought himself safe for the     
session1 and now found himself subjected
 to a heckling which he probably        
expected would be repeated indefinitely1
 may be imagined4  ,the subject of the  
first lecture which the writer attended 
was simple harmonic motion1 and was     
illustrated by means of pendulums1      
spiral springs with weights1 a long     
vertical rod of steel tipped with an    
ivory ball and fastened to a heavy base1
 tuning-forks1 etc4  ,the motion was    
defined as that of a particle moving    
along the diameter of a circle--the     
auxiliary circle1 ,thomson called it--so

 as always to keep pace1 as regards     
displacement in the direction along that
 diameter1 with a particle moving with  
uniform speed in the circle4 ,then the  
velocity and acceleration were found1   
and it was shown that the particle was  
continually accelerated towards the     
centre in proportion to the distance of 
the particle from that point4 ,the      
constant ratio of acceleration to       
displacement was proved to be equal to  
the square of the angular velocity in   
the auxiliary circle1 and from this     
fact1 and the particular value of the   
acceleration when the particle was at   
either end of its range of motion1 an   
expression for the period in terms of   
the speed and radius of the auxiliary   
circle was deduced4 ,then the ordinary  
simple pendulum formula was obtained4   
,this mode of treatment of an elementary
 matter1 so entirely different from     
anything in the ordinary text-books1    
arrested the attention at once1 and     
conveyed1 to some at least of those     

present1 an idea of simple harmonic     
motion which was directly applicable to 
all kinds of cases1 such as the motion  
of the air in a sound wave1 or of the   
medium which conveys the waves of light4
  ,the subject of ,kepler's laws was    
dealt with in the early lectures of     
every course1 and ,newton's deductions  
were insisted on as containing the      
philosophy of the whole question1       
leading1 as they did1 to the single     
principle from which the laws could be  
deduced1 and the third law corrected    
when the mass of the planet was         
comparable with that of the sun4        
,sometimes ,thomson would read the      
remarkable passage in ,hegel's ,logik1  
in which he refers to the ,newtonian    
theory of gravitation and says1 ,the    
planets are not pulled this way and     
that1 they move along in their orbits   
like the blessed gods1 and remark upon  
it4 ,on one occasion his remark was1    
,well1 gentlemen1 if these be his       
physics1 what must his metaphysics be8  

,and certainly that a philosopher should
 deny1 as ,hegel seemed to do1 all merit
 to the philosophical setting in which  
,newton placed the empirical results of 
,kepler1 is a very remarkable           
phenomenon4  ,the vivacity and          
enthusiasm of the ,professor at that    
time were very great4 ,the animation of 
his countenance as he looked at a       
gyrostat spinning1 standing on a        
knife-edge on the glass plate in front  
of him1 and leaning over so that its    
centre of gravity was on one side of the
 point of support2 the delight with     
which he showed that hurrying of the    
precessional motion caused the gyrostat 
to rise1 and retarding the precessional 
motion caused the gyrostat to fall1 so  
that the freedom to precess was the     
secret of its not falling2 the immediate
 application of the study of the        
gyrostat to the explanation of the      
precession of the equinoxes1 and        
illustration by a model of a terrestrial
 globe1 arranged so that the centre     

should be a fixed point1 while its      
axis--a material spike of brass--rolled 
round a horizontal circle1 the centre of
 which represented the pole of the      
ecliptic1 and the diameter of which     
subtended an angle at the centre of the 
globe of twice the obliquity of the     
ecliptic2 the pleasure with which he    
pointed to the motion of the equinoctial
 points along a circle surrounding the  
globe on a level with its centre1 and   
representing the plane of the ecliptic1 
and the smile with which he announced1  
when the axis had rolled once round the 
circle1 that #bf1jjj years had          
elapsed--all these delighted his        
hearers1 and made the lecture memorable4
  ,then the gyrostat1 mounted with its  
axis vertical on trunnions on a level   
with the fly-wheel1 and resting on a    
wooden frame carried about by the       
professor6 ,the delight of the students 
with the quiescence of the gyrostat when
 the frame1 gyrostat and all1 was       
carried round in the direction of the   

spin of the fly-wheel1 and its sudden   
turning upside down when the frame was  
carried round the other way1 was        
extreme1 and when he suggested that a   
gyrostat might be concealed on a tray of
 glasses carried by a waiter1 their     
appreciation of what would happen was   
shown by laughter and a tumult of       
applause4  ,some would have liked to    
follow the motions of spinning bodies a 
little more closely1 and to have made   
out clearly why they behaved as they    
did4 ,apparently ,thomson imagined the  
whole affair was self-evident1 for he   
never gave more than the simple         
parallelogram diagram showing the       
composition1 with the already existing  
angular momentum about the axis of the  
top1 of that generated about another    
axis1 in any short time1 by the action  
of gravity4  ,as a matter of fact1 the  
stability and instability of the        
gyrostat on the tray give the best      
possible illustration of the two        
different forms of solution of the      

differential equation1 3    #j1         
according as  is positive or negative2  
though it is also possible to explain   
the inversion very simply from first    
principles4 ,all this was no doubt      
regarded by ,thomson as obvious2 but it 
was far from being self-evident to even 
good students of the ordinary class1    
who1 without exception1 were beginning  
the study of dynamics4  ,thomson's      
absorption in the work of the moment was
 often very great1 and on these         
occasions he much disliked to be brought
 down to sublunary things by any slight 
mischance or inconvenience4 ,examples   
will occur to every old pupil of the    
great emphasis with which he commanded  
that precautions should be taken to     
prevent the like from happening again4  
,copies of ,thomson and ,tait's ,natural
 ,philosophy--,t and ,t' was its        
familiar title--and of other books1     
including ,barlow's ,tables and other   
collections of numerical data1 were     
always kept on the lecture-table4 ,but  

occasionally a laboratory student would 
stray in after everything had been      
prepared for the morning lecture1 and   
carry off ,barlow to make some          
calculation1 and of course forget to    
return it4 ,next morning some number    
would be wanted from ,barlow in a hurry1
 and the book would be missing4 ,then   
,thomson would order that ,barlow should
 be chained to the lecture-table1 and   
enjoin his assistant to see that that   
was done without an hour's delay6  ,on  
one occasion1 after working out part of 
a calculation on the long fixed         
blackboard on the wall behind the table1
 his chalk gave out1 and he dropped his 
hand down to the long ledge which       
projected from the bottom of the board  
to find another piece4 ,none was just   
there2 and he had to walk a step or two 
to obtain one4 ,so he enjoined          
,mc,farlane1 his assistant1 who was     
always in attendance1 to have a         
sufficient number of pieces on the ledge
 in future1 to enable him to find one   

handy wherever he might need it4        
,mc,farlane forgot the injunction1 or   
could not obtain more chalk at the time1
 and the same thing happened next day4  
,so the command was issued1 ,mc,farlane1
 ,i told you to get plenty of chalk1 and
 you haven't done it4 ,now have a       
hundred pieces of chalk on this ledge   
to-morrow2 remember1 a hundred pieces2  
,i will count them6 ,mc,farlane1 afraid 
to be caught napping again1 sent that   
afternoon for several boxes of chalk1   
and carefully laid the new shining white
 sticks on the shelf1 all neatly        
parallel at an angle to the edge4 ,the  
shelf was about sixteen feet long1 so   
that there was one piece of chalk for   
every two inches1 and the effect was    
very fine4 ,the class next morning was  
delighted1 and very appreciative of     
,mc,farlane's diligence4 ,thomson came  
in1 put up his eye-glass1 looked at the 
display1 smiled sweetly1 and1 turning to
 the applauding students1 began his     
lecture4  ,from time to time there were 

special experiments1 which excited the  
interest of the class to an             
extraordinary degree4 ,one was the      
determination of the velocity of a      
bullet fired from a rifle into a ,robins
 ballistic pendulum4 ,the pendulum1     
consisting of a massive bob of lead     
attached to a rigid frame of iron bars  
turning about knife-edges1 was set up   
behind the lecture-table1 and the bullet
 was fired by ,thomson from a ,jacob    
rifle into the bob of the pendulum4 ,the
 velocity was deduced from the          
deflection of the pendulum1 its known   
moment of inertia about the line of the 
knife-edges1 the distance of the line of
 fire from that line1 and the mass of   
the bullet4  ,in some of the notices of 
,lord ,kelvin that have appeared in the 
newspapers1 the imagination of the      
writers has converted the ,jacob rifle  
into one which ,professor ,thomson      
carried in the early years of the       
volunteer movement1 as a member of a    
,glasgow corps4 ,it is still used in the

 ,natural ,philosophy ,department for   
the same experiment1 and is a           
muzzle-loading rifle of large calibre1  
which throws an ounce bullet4 ,it was   
invented by the well-known ,indian      
sportsman1 ,colonel ,jacob1 for big-game
 shooting in ,india4 ,thomson held a    
commission as captain in the ,k 7or     
,university7 ,company of rifle          
volunteers1 and so did not shoulder a   
rifle1 except when he may have indulged 
in target practice4  ,the front bench   
students were always in a state of      
excitement1 mingled in some cases       
perhaps with a little trepidation4 ,for 
the target was very near them1 and      
though danger was averted by placing a  
large wooden screen in front of the bob1
 to prevent splinters of the bullet from
 flying about in the event of its       
missing the target and striking the iron
 casing of the bob1 there was a slight  
amount of nervousness as to what might  
happen4 ,the rifle1 loaded by           
,mc,farlane1 who had weighed out the    

charge of powder 7so many drams7 from a 
prescription kept in a cavity of the    
stock1 was placed on the table1 and two 
rests1 provided with  notches to receive
 the rifle1 were placed in the proper   
position to enable a bull's eye to be   
obtained4 ,thomson generally produced a 
small box of cotton wool1 and inserted a
 little in each of his ears to prevent  
injury to the tympanum from the report1 
and advised the spectators to do the    
same4 ,then1 adjusting his eye-glass1 he
 bent down1 placed the rifle in         
position1 and fired1 and the solemn     
stillness with which the aiming and     
adjustments had been witnessed was      
succeeded by vociferous applause4 ,the  
length of tape drawn out under a light  
spring was read off by ,mc,farlane1 who 
had already placed on the blackboard the
 formula for calculation of the         
velocity1 with the factor by which the  
length of tape had to be multiplied to  
give the velocity in feet per second4   
,then1 with the intimation that a       

question involving numerical calculation
 would be set on the subject1 in the    
ensuing ,monday morning examination     
paper1 the lecture generally closed1 or 
was rounded off with some further       
observations on angular 7or1 as ,thomson
 always preferred to call it1 moment of7
 momentum4  ,long after in the course of
 a debate in the ,house of ,lords on a  
proposal to make the use of the metric  
system of weights and measures          
compulsory1 ,lord ,kelvin told their    
lordships how he had weighed out the    
powder to charge this rifle1 and1       
mistaking the weights1 had loaded the   
rifle with an amount of powder which    
would have been almost certain to burst 
the piece1 but had happily paused before
 firing it off4  ,he often interrupted  
the course of a lecture with a          
denunciation of the ,british no-system  
of weights and measures--insane1        
brain-wasting1 dangerous1 were among the
 mildest epithets he applied to it1 and 
he would deeply sympathise with the     

student whose recollection of           
avoirdupois weight1 troy weight1        
apothecaries' weight1 etc41 was somewhat
 hazy4 ,the danger of the system        
consisted mainly in the fact that the   
apothecaries' dram is #fj grains1 while 
the avoirdupois dram is #bg grains4     
,thus so many drams of powder required  
to charge a rifle is a very much larger 
quantity when reckoned in apothecaries' 
drams than when reckoned in avoirdupois4
 ,as a rule he left the loading of the  
rifle1 like all the other lecture-room  
experiments1 to his assistants4         
,another experiment which caused a great
 sensation was that known as the        
dewdrop6 ,a funnel of brass1 composed of
 a tube about #cj inches long and an    
inch wide1 and a conical mouth about ten
 inches wide1 had a piece of stout sheet
 ,india-rubber stretched1 as tightly as 
it could be by hand1 across its mouth1  
and made water-tight by a serving of    
twine and cement round the edge4 ,a wire
 soldered round the outside of the lip  

gave a good hold for this serving and   
made all perfectly secure4 ,on the plane
 surface of the sheet geometrical       
figures were drawn in ink1 so that their
 distortion could be afterwards studied4
 ,the funnel was then hung by a strong  
support in an inverted position behind  
the table1 and water poured gently into 
it from a rubber supply pipe connected  
with the water-main4 ,as the water was  
allowed to accumulate--very slowly at   
first--the sheet of rubber gradually    
stretched and bulged out1 at first to a 
flat lens-shape1 and gradually more and 
more1 till an immense water-drop had    
been formed1 #ae or #ah inches in       
horizontal diameter1 and of still       
greater vertical dimensions4 ,the rubber
 film was now1 at the place of greatest 
tension1 quite thin and transparent1 and
 its giving way was anticipated by the  
students with keen enjoyment4 ,a large  
tub had been placed below to receive the
 water1 but the deluge always extended  
over the whole floor space behind the   

table1 and was greeted with rapturous   
applause4  ,before the drop burst1 and  
while it was forming1 ,thomson          
discoursed on surface tension1          
emphasising the essential difference    
between the tension in the rubber-film  
and the surface-film of a dewdrop1 and  
pointing out how the geometrical figures
 had changed in shape4 ,then he would   
poke it with the pointer he held in his 
hand1 and1 turning to the class1 as the 
mass quivered1 remark1 ,the trembling of
 the dewdrop1 gentlemen6  ,vibrations of
 elastic solids were illustrated in     
various ways1 frequently by means of a  
symmetrical shape of calves'-foot jelly1
 at the top of which a coloured marble  
had been imbedded as a molecule1 the    
motions of which could be followed4 ,and
 then he would discourse on the         
,poisson-,navier theory of isotropic    
solids1 and the impossibility of the    
fixed relation which that theory imposed
 between the modulus of rigidity and the
 modulus of compression2 and refer with 

approval to the series of examples of   
perfectly uniform1 homogeneous1         
isotropic solids1 which ,stokes had     
shown could be obtained by making       
jellies of different degrees of         
stiffness4 ,another example1 frequently 
adduced as indicating the falsity of the
 theory1 was the entirely different     
behaviour of blocks of ,india-rubber and
 cork1 under compression applied by a   
,bramah press4 ,the cork diminished in  
thickness without spreading out         
laterally2 the rubber1 being very little
 compressible1 bulged out all round as  
its thickness was diminished4  ,the     
lectures on acoustics1 which came late  
in the course1 were also exceedingly    
popular4 ,two ,french horns1 with all   
their crooks and accessories1 were      
displayed1 and sometimes1 to the great  
delight of the class1 ,thomson would    
essay to show how the pitch of a note   
could be modified by means of the keys1 
or by the hand inserted in the bell4    
,the determination by the siren of the  

pitch of the notes of tuning-forks      
excited by a 'cello bow1 and the tuning 
of a major third by sounding at the same
 time the perfect fifth of the lower    
note1 were often exhibited1 and         
commented on with acute remarks1 of     
which it is a pity no statement was ever
 published4#be  ,the closing lecture of 
the ordinary course was usually on      
light1 and the subject which was        
generally the last to be taken up--for  
as the days lengthened in spring1 it was
 possible sometimes to obtain sunlight  
for the experiments--was often relegated
 to the last day or two of the session4 
,so after an hour's lecture ,thomson    
would say1 ,as this is the last day of  
the session1 ,i will go on for a little 
longer1 after those who have to leave   
have gone to their classes4 ,then he    
would resume after ten o'clock1 and go  
on to eleven1 when another opportunity  
would be given for students to leave1   
and the lecture would be again resumed4 
,messengers would be sent from his      

house1 where he was wanted for business 
of different sorts1 to find out what had
 become of him1 and the answer brought  
would be1 hour after hour1 ,he is still 
lecturing4 ,at last he would conclude   
about one o'clock1 and gently thank the 
small and devoted band who had remained 
to the end1 for their kind and prolonged
 attention4  ,in the course of his      
lectures ,thomson continually called on 
his assistants for data of all kinds4   
,in the busiest time of his life--the   
fifteen years from #ahgj to #ahhe--;he  
trusted to his assistants for the       
preparation of his class illustrations1 
and it was sometimes a little difficult 
to anticipate his wishes1 for without   
careful rehearsal it is almost          
impossible to make sure that in an      
experimental lecture everything will go 
without a hitch4 ,the digressions1      
generally most interesting and          
instructive1 in which he frequently     
indulged1 almost always rendered it     
necessary to bring some experiment      

before the class which had not been     
anticipated1 and all kinds of things    
were kept in readiness1 lest they should
 be wanted suddenly4  ,it has often been
 asserted that ,thomson appealed to his 
assistant for information contained in  
the multiplication-table1 and could not 
perform the ordinary operations of      
arithmetic4 ,his active mind1 working on
 ahead of the statements he was making  
at the moment1 often could not be       
brought back to the consideration of the
 value of #i times #f1 and the like2 but
 it was quite untrue that he was        
incapable of making calculations4 ,his  
memory was good1 and though he never    
could be1 for example1 sure whether the 
aqueous humour was before or behind the 
crystalline in the eye1 he was generally
 able at once to tell when a            
misstatement had been made as to any    
numerical question regarding the subject
 under discussion4  ,in the higher      
mathematical class1 to which he lectured
 on ,wednesdays1 at noon1 ,thomson was  

exceedingly interesting4 ,there he      
seemed to work at the subject as he     
lectured2 new points to be investigated 
continually presented themselves1 and   
the students were encouraged to work    
them out in the week-long intervals     
between his lectures4 ,always the       
physical interpretation of results was  
aimed at1 even intermediate steps were  
discussed4 ,thus the meaning of the     
mathematical processes was ever kept in 
view1 and the men who could follow were 
made to think while they worked1 and to 
regard the mathematical analysis as     
merely an aid1 not an end in itself4 ,a 
little expenditure of chalk is a saving 
of brains2 the art of reading           
mathematical books is judicious         
skipping1 were remarks he sometimes     
made1 and illustrate his view of the    
relative importance of mathematical work
 when he regarded it as the handmaid of 
the physical thinker4 ,yet he valued    
mathematics for its own sake1 and was   
keenly alive to elegance of form and    

method1 as readers of such great        
mathematical discussions as the         
,appendix on ,spherical ,harmonics1 in  
,thomson and ,tait1 will observe4 ,he   
spoke with unqualified admiration of the
 work of ,green and ,stokes1 of         
,cauchy's great memoir on ,waves1 and of
 ,hamilton's papers on ,dynamics4 ,but  
no form of vector-analysis1 neither the 
,quaternions of ,hamilton nor the       
,vectors of ,willard ,gibbs and         
,heaviside1 appealed to him1 and the    
example of his friend and co-worker1    
,tait1 had no effect in modifying his   
adverse verdict regarding this          
department of mathematics1 a verdict    
which in later years became only more   
emphatic4  ,one session he began the    
first lecture of the higher class by    
writing dxdt in the middle of the       
blackboard1 and demanding of each of the
 ten or a dozen students present1 some  
of them distinguished graduates1 what it
 meant6 ,one student described it as the
 limiting value of the ratio of the     

increment of the dependent variable x to
 the increment of the independent       
variable t1 when the latter increment is
 made indefinitely small4 ,he retorted1 
,that's what ,todhunter would say6 ,the 
others gave various slightly different  
versions of the same definition4 ,at    
last he impatiently remarked1 ,does     
nobody know that dxdt means velocity8   
,here the physical idea as a whole was  
before his mind2 and he did not reflect 
that if t denoted time and x distance in
 any direction1 the explanation given by
 the student did describe velocity with 
fair accuracy4  ,an embarrassing        
peculiarity of his mathematical         
discussions was his tendency1 when a    
difficulty of symbolisation occurred1 to
 completely change the notation4 ,also  
he was not uniformly accurate in        
analytical work2 but he more than made  
up for this by the faculty he had of    
devising a test of the accuracy of the  
result and of divining the error which  
had crept in1 if the test was not       

satisfied4  ,the subjects he treated    
were always such great branches of      
mathematics as the theory of the        
tides--he discussed the tidal phenomena 
of the ,english ,channel in one         
course--the general theory of           
vibrations1 ,fourier analysis1 the      
theory of waves in water1 etc41 etc4 ,a 
very good idea of the manner and matter 
of his mathematical prelections can be  
obtained from a perusal of the          
,baltimore ,lectures4  ,in the physical 
laboratory he was both inspiring and    
distracting4 ,he continually thought of 
new things to be tried1 and interrupted 
the course of the work with interpolated
 experiments which often robbed the     
preceding sequence of operations of     
their final result4 ,his ideas were on  
the whole better worked out by a really 
good corps of students when he was from 
home1 and could only communicate by     
letter his views on the work set forth  
in the daily reports which were         
forwarded to him4  ,he insisted with    

emphasis that a student who found that a
 quadrant electrometer would not work   
well should take it to pieces to        
ascertain what was the matter4 ,this of 
course generally resulted in the return 
of the instrument to ,white's shop to be
 put together again and adjusted4 ,but1 
as he said1 there was a cause for every 
trouble of that kind1 and the great     
thing was to find out at once what it   
was4  ,thomson's concentration on the   
work in hand1 and his power of simply   
taking possession of men1 even mere     
spectators1 and converting them into    
assistants1 was often shown in the      
laboratory4 ,several men who have since 
become eminent were among the assistants
 enrolled from the laboratory students4 
,professor ,w4 ,e4 ,ayrton and1 later1  
,professor ,john ,perry1 were students  
at ,glasgow for a time1 and rendered the
 most able and willing help in the      
researches which were then proceeding4  
,this power was1 no doubt1 the secret of
 his success in gathering round him an  

enthusiastic corps of laboratory workers
 in the early years of his              
professorship1 and it was shown also by 
the ease with which he annexed the      
,blackstone examination-room and1 later1
 various spaces in the new ,university  
buildings4 ,there1 after a time1 the    
,natural ,philosophy rooms were found by
 the senatus to include not only the    
original class-room1 laboratory1 etc41  
but also all the spare attics and       
corridors in the neighbourhood1 and even
 the ,university tower itself6 ,one of  
his colleagues1 who venerated him       
highly1 remarked recently1 ,he had a    
great faculty for annexation6  ,the     
incident referred to occurred while he  
was preparing the article on ,heat for  
the ninth edition of the ,encyclopdia   
,britannica4 ,it seemed at first a pity 
that ,thomson should undertake to write 
such articles2 but in the course of     
their preparation he came upon so many  
points on which experimental information
 was wanting1 and instituted so many    

researches to answer his questions1 that
 the essays took very much the character
 of original papers4 ,in the article on 
,heat 7he also wrote ,elasticity71 will 
be found a long account of ,steam       
,thermometry1 that is1 of thermometers  
in which the indicating substance was to
 be the saturated vapours of different  
substances1 water1 sulphurous acid1     
etc41 etc41 for he did not limit the    
term steam to water-vapour4 ,for some   
time every one in the laboratory was    
employed in making sulphurous acid1 by  
heating copper in sulphuric acid in the 
usual way1 and condensing the gas in    
tubes immersed in freezing mixtures2 and
 the atmosphere of the room was of a    
sort which1 however noxious to germs of 
different kinds1 it was a little        
difficult to breathe4 ,one morning1 when
 all were thus occupied1 an eminent     
chemist1 who had just come home from the
 south for a vacation1 called to pay his
 respects4 ,after a word or two of      
inquiry as to how his young friend was  

prospering in his new post1 ,thomson    
said1 ,we are all very busy brewing     
liquid sulphurous acid1 for use in      
sulphurous acid steam thermometers2 we  
want a large quantity of the liquid2    
would you mind helping us8 ,so1 desiring
 an assistant to find a flask and       
materials1 he enrolled this new and     
excellent recruit on the spot2 and what 
was intended to be a mere call1 was     
prolonged into a long day of ungrudging 
work at an elementary chemical exercise6
     ,,chapter ,,xvii  ,,practical      
,,activities--,,honours ,,and           
,,distinctions--,,last ,,illness ,,and  
,,death   ,it remains to say something  
of ,lord ,kelvin's public and practical 
activities4 ,all over the world he came 
ultimately to be recognised as the      
greatest living scientific authority in 
almost all branches of physics4 ,every  
existing learned society sought to make 
him a ,fellow1 honorary degrees were    
showered on him from all quarters4 ,a   
list of some of the most important of   

these distinctions is given in the      
,royal ,society ,year-,book for #aijg2  
it is doubtful if a complete list could 
be compiled4 ,he was awarded the ,keith 
,medal and the ,victoria ,jubilee ,medal
 by the ,royal ,society of ,edinburgh1  
and received in succession the ,copley  
and ,royal ,medals of the ,royal        
,society of ,london1 of which he was    
elected a ,fellow in #ahea1 and was     
,president from #ahij to #ahie4 ,for    
several periods of years he was         
,president of the ,royal ,society of    
,edinburgh1 to which he communicated his
 papers on heat1 dissipation of energy1 
vortex motion1 and many other memoirs4  
,he was ,president of the ,british      
,association at the ,edinburgh meeting  
in #ahga1 when he delivered a           
presidential address1 noteworthy in many
 respects1 but chiefly remarkable in the
 popular mind on account of his         
suggestion that life was conveyed to the
 earth by a seed1 a germ enclosed in a  
crevice of a meteorite4 ,this was       

understood at the time by many people as
 an attempt to explain the origin of    
life itself1 instead of what it was     
intended to be1 an explanation of the   
beginning of the existence of living    
things on a planet which was originally1
 on the completion of its formation by  
the condensation of nebular matter1 red 
hot even at its surface4 ,on several    
occasions he was president of ,section  
,a1 and he was constant in attendance at
 the ,association meetings1 and an eager
 listener and participator in the       
discussions and debates4 ,his scientific
 curiosity was never at rest1 and he    
dearly liked to meet and converse with  
scientific workers4  ,lady ,thomson1 who
 had been long an invalid1 died in      
#ahgj1 and in #ahgd ,sir ,william       
,thomson was married to ,miss ,frances  
,anna ,blandy 7daughter of ,mr4 ,charles
 ,r4 ,blandy of ,madeira7 who survives  
him as ,lady ,kelvin4 ,to her tender    
solicitude he owed much of his constant 
and long-continued activity in all kinds

 of work4 ,she accompanied him on all   
public occasions1 and he relied greatly 
on her helpfulness and ever watchful    
care4  ,in #ahib ,sir ,william ,thomson1
 while ,president of the ,royal         
,society1 was raised to the ,peerage1   
with the title of ,baron ,kelvin of     
,netherhall1 ,largs2 and more lately he 
was created a member of the ,order of   
,merit and a ,g4,c4,v4,o4 ,his foreign  
distinctions were very numerous4 ,he was
 a ,knight of the ,order ,pour le ,mrite
 of ,prussia1 a ,foreign ,associate of  
the ,institute of ,france1 and a ,grand 
,officer of the ,legion of ,honour4 ,but
 no public honour or mark of royal      
favour could raise him in the estimation
 of all who know anything of science or 
of the labours of the scientific men to 
whom we owe the necessities and luxuries
 of our present civilisation4  ,in #ahif
 the ,city and ,university of ,glasgow  
celebrated the jubilee of his           
,professorship of ,natural ,philosophy4 
,the rejoicings on that occasion will   

never be forgotten by those whose       
privilege it was to take part in them4  
,delegates came from every country in   
the world1 and kings and princes1       
universities and learned societies1     
colleges and scholastic institutions of 
every kind1 vied with each other in     
doing honour to the veteran who had     
fought for truth and light for so many  
years1 and won so many victories4 ,a    
memorial volume of the proceedings was  
published1 including a review of ,lord  
,kelvin's work by the late ,professor   
,fitz,gerald1 and a full report appeared
 in ,nature and other journals at the   
time1 so that it is unnecessary to give 
particulars here4 ,and indeed it is     
impossible by any verbal description to 
convey an idea of the enthusiasm with   
which the scientific world acclaimed its
 leader1 and of the dignity and state of
 the ceremonies4  ,in #ahii1 at the age 
of seventy-five1 ,lord ,kelvin resigned 
the ,chair of ,natural ,philosophy1 and 
retired1 not to rest1 but to investigate

 more vigorously than ever the          
properties of matter4 ,one remarkable   
fruit of his leisure we have in his     
great book1 the ,baltimore ,lectures1 in
 which theories of light are discussed  
with a power which excites the reverence
 of all engaged in the new researches   
and which recent discoveries have called
 into existence4 ,and it is not too much
 to say that the means of discussing and
 extending these discoveries are in     
great measure due to ,lord ,kelvin4     
,during the year #aijg ,lord ,kelvin    
performed many ,university duties and   
seemed to be in unusually good health4  
,he presided as ,chancellor at the      
installation of ,mr4 ,asquith as ,lord  
,rector on ,january #aa1 and in the same
 capacity attended a few days later the 
funeral of ,principal ,story1 the       
,vice-,chancellor1 who died on ,january 
#ac4 ,on ,april #bc he presided at the  
long and arduous ceremonies of honorary 
graduation1 and the public opening of   
the new ,natural ,philosophy ,institute 

and the new ,medical ,buildings1 by the 
,prince of ,wales4 ,as ,chancellor he   
conferred the degree of ,doctor of ,laws
 on the ,prince and ,princess1 and took 
the chair at the luncheon which followed
 the proceedings1 when he proposed in a 
short and graceful speech the health of 
the ,princess4  ,he was able to take    
part also in various political and      
social meetings1 and to give attention  
to the work in progress at the factories
 of his firm in ,cambridge ,street4     
,lady ,kelvin and he left ,netherhall1  
,largs1 for ,aix les ,bains1 at the end 
of ,july1 but visited the ,british      
,association at ,leicester in passing4  
,there he heard the presidential address
 of his old friend1 ,sir ,david ,gill1  
to whom he moved a vote of thanks in his
 usual vivacious manner4  ,lord ,kelvin 
had been accustomed for a good many     
years to spend a month or six weeks in  
summer or early autumn at the famous    
,french watering-place1 from which he   
seemed always to receive much benefit4  

,for a long time he had suffered from an
 intermittent and painful form of facial
 neuralgia1 which1 except during its    
attacks1 which came and passed suddenly1
 did not incapacitate him from work4    
,with the exception of a rather serious 
illness in #aijf1 this was the only     
ailment from which he had suffered for  
many years1 and his general health was  
otherwise uniformly good4  ,lord and    
,lady ,kelvin returned to ,netherhall on
 ,september #ad1 with the intention of  
going in a day or two to ,belfast1 to   
open the new scientific buildings of    
,queen's ,college4 ,but1 unfortunately1 
on the day of their arrival ,lady       
,kelvin became very seriously ill1 and  
the visit to ,ireland had to be         
abandoned4 ,his address was1 however1   
read by his nephew1 ,james ,thomson1 son
 of his elder brother1 and was a tribute
 to the city of his birth1 and the      
memory of his father4  ,the illness of  
,lady ,kelvin caused much anxiety for   
many weeks1 and this1 and perhaps some  

incautious exposure1 led to the         
impairment of ,lord ,kelvin's health4 ,a
 chill caught on ,november #bc caused   
him to be confined to bed2 and though he
 managed for a week or two still to do  
some work on a paper with which he had  
been occupied for a considerable time1  
he became worse1 and gradually sank1    
until his death at a quarter-past ten   
o'clock on the evening of ,december #ah4
  ,the keen sorrow which was universally
 felt for ,lord ,kelvin's death was     
manifested by all classes of the        
community4 ,in ,glasgow every one       
mourned as for the greatest of the land1
 and the testimony to the affection in  
which he was held1 and the reverence for
 his character and scientific           
achievements1 was extraordinary4 ,and   
this feeling was universal2 from all    
parts of the world poured in telegrams  
of respectful sympathy with ,lady       
,kelvin and with the ,university of     
,glasgow in their bereavement4  ,the    
view was immediately and strongly       

expressed1 both privately and by the    
press1 that the most illustrious natural
 philosopher since ,newton should rest  
beside the great founder of physical    
science in ,westminster ,abbey1 and a   
requisition was immediately prepared and
 forwarded by the ,royal ,society of    
,london to the ,dean of ,westminster4   
,the wish of the whole scientific world 
was at once acceded to1 and on ,december
 #bc1 at noon1 the interment took place1
 with a state and yet a simplicity which
 will never be forgotten by those who   
were present4  ,nearly all the          
scientific notabilities of the country  
were present1 and the coffin1 preceded  
by the choristers and the clergy1 while 
the hymn1 ,brief life is here our       
portion1 was sung1 was followed round   
the cloistered aisles from ,st4 ,faith's
 chapel to the choir1 by the relatives1 
representatives of ,his ,majesty the    
,king and the ,prince of ,wales1 by the 
,royal ,society1 by delegates from the  
,institute of ,france1 representatives  

of the ,universities of ,cambridge1     
,oxford1 ,glasgow1 and other            
universities1 of the ,royal ,society of 
,edinburgh 7of which ,lord ,kelvin was  
president when he died71 and of most of 
the learned societies of the kingdom4   
,then1 after a short service1 the body  
was followed to the grave in the        
cloisters by the same company of        
mourners1 and to the solemn words of the
 ,burial ,service was laid close by     
where rests all that was mortal of      
,isaac ,newton4 ,there he sleeps well   
who toiled during a long life for the   
cause of natural knowledge1 and served  
nobly1 as a hero of peace1 his country  
and the world4     ,,conclusion   ,the  
imperfect sketch of ,lord ,kelvin's     
scientific life and work which this book
 contains can only give a faint notion  
of the great achievements of the long   
life that has now ended4 ,beyond the    
researches which he carried out and the 
discoveries he made1 there is the       
inspiration which his work and example  

gave to others4 ,inspired himself by    
,lagrange1 ,laplace1 ,ampre1 and        
,fourier1 and led to experimental       
research by the necessity for answers to
 the questions which his mathematical   
expression of the discoveries of the    
twenty-five years which preceded the    
establishment of his laboratory had     
suggested--the theories of electricity  
and magnetism1 of heat1 of elasticity1  
his discoveries in general dynamics and 
in fluid motion1 the publication of     
,thomson and ,tait1 all made him the    
inspirer of others2 and there was no    
one1 however eminent1 who was not proud 
to acknowledge his obligations to his   
genius4 ,clerk ,maxwell1 before he wrote
 the most original treatise on          
electricity that has ever appeared1 gave
 himself to the study of ,faraday's     
,experimental ,researches and to the    
papers of ,thomson4 ,and if some1 like  
,fitz,gerald and others1 have regretted 
that the electromagnetic theory of light
 to which ,maxwell was led by ,faraday1 

and1 indeed1 by ,thomson himself1 did   
not meet with a more sympathetic        
reception at his hands1 they have not   
been unmindful of the source from which 
much of their illumination has come4    
,he has founded a school of thought in  
mathematical physics1 of men in whose   
minds the symbol is always the servant  
of the ideas1 whose motto is            
interpretation by dynamical processes   
and models as far as that is possible1  
who shirk no mathematical difficulties  
when they have to be encountered1 but   
are never led away from the straight    
road to the goal which they seek to     
reach--the systematic and clear         
formulation of the course of physical   
action4  ,and in ,lord ,kelvin's mind   
there was blended with a clear physical 
instinct which put aside all that was   
extraneous and unessential to the main  
issue an extraordinary power of         
concentration on the problem in hand1   
and a determination that was never      
daunted by failure1 which consented to  

postponement but never to               
relinquishment1 and which led often     
after long intervals of time to success 
in the end4 ,he believed that light     
would come at last on the most baffling 
of problems1 if only it were looked at  
from every point of view and its        
conditions were completely formulated2  
but he could put what was for the time  
impossible aside1 and devote himself to 
the immediately possible and realisable4
 ,and as often happens with every       
thinker1 his mind1 released from the    
task1 returned to it of itself1 and what
 before appeared shrouded in            
impenetrable mist stood out suddenly    
sharp and distinct like a mountain-top  
before a climber who has at last risen  
above the clouds4  ,with the great      
mathematical power and sure instinct    
which led him to success in physical    
research was combined a keen perception 
of the importance of practical          
applications4 ,sometimes the practical  
question suggested the theoretical and  

experimental research1 as when the needs
 of submarine telegraphy led to the     
discussion of the speed of signalling   
and the evolution of the reflecting     
galvanometer and the siphon recorder4   
,on the other hand1 the mathematical    
theory of electricity and magnetism had 
led to quantitative measurement and     
absolute units at an earlier time1 when 
the need for these was beginning to be  
felt clearly by scientific workers and  
dimly by those far-sighted practical men
 who dreamed--for a dream it was thought
 at the time--of linking the ,old ,world
 with the ,new by a submarine cable4    
,but the quantitative study of          
electricity in the laboratory threw     
light on economic conditions1 and the   
mass of data already obtained1 mainly as
 a mere matter of experimental          
investigation of the properties of      
matter1 became at once a valuable asset 
of the race of submarine cable engineers
 which suddenly sprang into existence4  
,and so it has been with the more recent

 applications of electricity4 ,the      
induction of currents discovered by     
,faraday could not become of practical  
importance until its laws had been      
quantitatively discussed1 a much longer 
process than that of discovery2 and we  
have seen how the ,british ,association 
,committee1 led by ,thomson and         
,maxwell1 brought the ideas and         
quantities of this new branch of science
 into numerical relation with the units 
of already existing practical           
enterprise4 ,the electrical measuring   
instruments--first the electrometers1   
and more recently the electric current  
balances and other beautiful instruments
 for the dynamo-room and the            
workshop--which ,lord ,kelvin invented  
have brought the precision of the       
laboratory into the everyday duties of  
the secondary battery attendant and the 
wireman4  ,and as to methods of         
measurement1 those who remember the     
haziness of even telegraph engineers as 
to the measurement of the efficiency of 

electrical currents and electromotive   
forces in the circuits of lamps and     
dynamos1 in the early days of electric  
lighting1 know how much the world is    
indebted to ,thomson4#bf ,he it was who 
showed at first how cables were to be   
tested1 as well as how they were to be  
worked2 it was his task1 again1 to show 
how instruments were to be calibrated   
for practical measurement of current and
 energy supplied by the early           
contractors to consumers4 ,he had in the
 quiet of his laboratory long before    
elaborated methods of comparing         
resistances1 and given the ,wheatstone  
balance its secondary conductors for the
 comparison of low resistances2 he now  
showed how the same principles could be 
applied to measure the efficiencies of  
dynamos and to make up the account of   
charge and discharge for a secondary    
battery4  ,and if the siphon-recorder   
and the mariners' compass and the       
sounding machine proved pecuniarily     
profitable1 the reward was that of the  

inventor1 who has an indefeasible right 
to the fruit of his brain and his hand4 
,but ,lord ,kelvin's activity was not   
confined merely to those practical      
things which have1 to use the ordinary  
phrase1 money in them2 he gave his time 
and energies freely to the perfecting of
 the harmonic analysis of the tides1    
undertook again1 for a ,committee of the
 ,british ,association1 the             
investigation of the tides for different
 parts of the world1 superintended the  
analysis of tidal records1 and invented 
tide-predicting machines and improved   
tide-gauges4  ,lord ,kelvin's work in   
the theory of heat and in the science of
 energy generally would have given him a
 title to immortality even if it had    
stood alone2 and there can be no doubt1 
even in the mind of the most determined 
practical contemner of the ,carnot      
cycle1 of the enormous importance of    
these achievements4 ,here he was a      
pioneer1 and yet his papers1 theoretical
 and yet practical1 written one after   

another in pencil and despatched1 rough 
as they were1 to be printed by the      
,royal ,society of ,edinburgh1 form1 as 
they are collected in volume i of his   
,mathematical and ,physical ,papers1 in 
some respects the best treatise on      
thermodynamics at the present time6     
,there are treatises written from a more
 general standpoint1 which deal with    
complex problems of chemical and        
physical change of means of             
thermodynamic potentials1 and processes 
which are not to be found set forth in  
this volume of papers2 but even these   
are to a great extent an outcome of his 
,thermoelastic1 ,thermomagnetic and     
,thermoelectric ,properties of ,matter4 
,in hydrodynamics also ,lord ,kelvin    
never lost sight of practical           
applications1 even while pursuing the   
most intensely theoretical researches   
into the action of vortices or the      
propagation of waves4 ,in his later     
years he worked out the theory of       
ship-waves with a power which has made  

more than one skilful and successful    
cultivator of this branch of science say
 that he was no mere mathematician1 but 
a man who1 like the prophets of old1    
could divine what is hid from the eyes  
of ordinary mortals4 ,of the ultimate   
importance of these for practical       
questions of the construction of ships1 
and the economy of fuel in their        
propulsion1 there can be little doubt4  
,unhappily1 the applications will have  
now to be made by others4  ,it is       
interesting to note that the            
investigation of waves in canals with   
which ,lord ,kelvin recently enriched   
the ,proceedings of the ,royal ,society 
of ,edinburgh have been carried out by a
 strikingly ingenious adaptation of the 
,fourier solution of the differential   
equation of the diffusion of heat along 
a bar1 or of electricity along a slowly 
worked cable4 ,thus1 beginning with     
,fourier mathematics in his earliest    
researches1 he has in some of his last  
work applied the special exponential    

form of ,fourier solution of the        
diffusion equation to a case1 that of   
wave propagation1 essentially different 
in physical nature1 and distinct in     
mathematical signification1 from that   
for which it was originally given4      
,lord ,kelvin's written work consists of
 the ,electrostatics and ,magnetism1    
three volumes of ,collected             
,mathematical and ,physical ,papers1    
three of ,popular ,lectures and         
,addresses1 the ,baltimore ,lectures1 a 
very considerable number of papers as   
yet uncollected1 and the ,natural       
,philosophy4 ,but this1 great as it was1
 represented only a relatively small    
part of his activities4 ,he advised     
public companies on special engineering 
and electrical questions1 served on     
,royal ,commissions1 acted as consulting
 engineer to cable companies and other  
corporations1 was employed as arbiter in
 disputes when scientific questions were
 involved1 advocated distinctive        
signalling for lighthouses and devised  

apparatus for this purpose1 and he was1 
above all1 a great inventor4 ,his       
patents are many and important4 ,one of 
them was for a water-tap warranted not  
to drip1 another1 for electrical        
generating machines1 meters1 etc41 was  
perhaps the patent of largest extent    
ever granted4  ,to ,lord ,kelvin's class
 teaching reference has been made in an 
earlier chapter4 ,he was certainly      
inspiring to the best students4 ,at     
meetings of the ,british ,association   
his luminous remarks in discussion      
helped and encouraged younger workers1  
and his enthusiasm was infectious4 ,but 
with the ordinary student who cannot    
receive or retain his mental nutriment  
except by a carefully studied mode of   
presentation1 he was not so successful4 
,he saw too much while he spoke2 new    
ideas or novel modes of viewing old ones
 presented themselves unexpectedly1     
associations crowded upon his mind1 and 
he was apt to be discursive1 to the     
perplexity of all except those whose    

minds were endued also with something of
 the same kind of physical instinct or  
perception4 ,then he was so busy with   
many things that he did not find time to
 ponder over and arrange the matter of  
his elementary lectures1 from the point 
of view of the presentment most suitable
 to the capacity of his hearers4 ,to the
 suggestion which has lately been made1 
that he should not have been obliged to 
lecture to elementary students1 he would
 have been the first to object4 ,as a   
matter of fact1 in his later years he   
lectured to the ordinary class only     
twice a week1 and to the higher class   
once4 ,the remainder of the lectures    
were given by his nephew1 ,dr4 ,j4 ,t4  
,bottomley1 who for nearly thirty years 
acted as his deputy as regards a great  
part of the routine work of the chair4  
,it is hardly worth while to refute the 
statement often made that ,lord ,kelvin 
could not perform the operations of     
simple arithmetic4 ,the truth is1 that  
in the class-room he was too eager in   

the anticipation of the results of a    
calculation1 or too busy with thoughts  
of what lay beyond1 to be troubled with 
the multiplication table1 and so he     
often appealed to his assistants for    
elementary information which at the     
moment his rapidly working mind could   
not be made to supply for itself4  ,to  
sum up1 ,lord ,kelvin's scientific      
activity had lasted for nearly seventy  
years4 ,he was born four years after    
,oersted made his famous discovery of   
the action of an electric current on a  
magnet1 and two years before ,ampre1    
founding on this experiment1 brought    
forth the first great memoir on         
electromagnetism4 ,thus his life had    
seen the growth of modern electrical    
science from its real infancy to its now
 vigorous youth4 ,the discoveries of    
,faraday in electrical induction were   
given to the world when ,lord ,kelvin   
was a boy1 and one of the great tasks   
which he accomplished was to weave these
 discoveries together in a uniform web  

of mathematical theory4 ,this theory    
suggested1 as we have seen1 new problems
 to be solved by experiment1 which he   
attacked with the aid of his students in
 the small and meagrely equipped        
laboratory established sixty years ago  
in the ,old ,college in the ,high       
,street4 ,it was his lot to live to see 
his presentations of theory lead to new 
developments in his own hands and the   
hands of other men of genius--,helmholtz
 and ,clerk ,maxwell1 for example--and  
to survive until these developments had 
led to practical applications throughout
 our industries1 and in all the affairs 
of present-day life and work4 ,his true 
monument will be his work and its       
results1 and to only a few men in the   
world's history has such a massive and  
majestic memorial been reared4  ,he was 
a tireless worker4 ,in every day of his 
life he was occupied with many things1  
but he was never cumbered4 ,the problems
 of nature were ever in his mind1 but he
 could put them aside in the press of   

affairs1 and take them up again         
immediately to push them forward another
 stage towards solution4 ,his green book
 was at hand on his table or in his     
pocket2 and whenever a moment's leisure 
occurred he had pencil in hand1 and was 
deep in triple integrals and            
applications of ,green's ,theorem1 that 
unfailing resource of physical          
mathematicians4    ,saepe stilum vertas 
quae digna legi sint   ,scripturus1  the
 motto which ,horace recommends1 was    
his1 and he would playfully quote it1   
pointing to the eraser-pad in the top of
 his gold pencil-case4 ,he erased1      
corrected1 amended1 and rewrote with    
unceasing diligence1 to the dismay of   
his shorthand-writing secretary4  ,the  
theories and facts of electricity and   
magnetism1 the production and           
propagation of waves in water or in the 
luminiferous ether1 the structure and   
density of the ether itself1 the        
relations of heat and work1 the motions 
of the heavenly bodies1 the constitution

 of crystals1 the theory of music1 the  
practical problems of navigation1 of    
telegraphing under the sea1 and of the  
electric lighting of cities--all these  
and more came before his mind in turn1  
and sometimes most of them in the course
 of a single day4 ,he could turn from   
one thing to another1 and find mental   
rest in diversity of mental occupation4 
,he would lecture from nine to ten      
o'clock in the morning to his ordinary  
class1 though generally this was by no  
means the first scientific work of the  
day4 ,at ten o'clock he passed through  
his laboratory and spoke to his         
laboratory students or to any one who   
might be waiting to consult him1        
answered some urgent letter1 or gave    
directions to his secretary2 then he    
walked or drove to ,white's workshop to 
immerse himself in the details of       
instrument construction until he was    
again due at the university for         
luncheon1 or to lecture to his higher   
mathematical class on some such subject 

as the theory of the tides or the       
,fourier analysis4  ,as scientific      
adviser to submarine telegraph companies
 and other public bodies1 and more      
recently as ,president of the ,royal    
,society of ,london1 he made frequent   
journeys to ,london4 ,these were        
arranged so as to involve the minimum   
expenditure of time4 ,he travelled by   
night when alone1 and could do so with  
comfort1 for he possessed the gift of   
being able to sleep well in almost any  
circumstances4 ,thus he would go to     
,london one night1 spend a busy day in  
all kinds of business--scientific1      
practical1 or political--and return the 
next night to ,glasgow1 fresh and eager 
for work on his arrival4 ,here may be   
noticed his power of detaching himself  
from his environment1 and of putting    
aside things which might well have been 
anxieties1 and of becoming again        
absorbed in the problem which           
circumstances had made him temporarily  
abandon4  ,genius has been said to be   

the power of taking infinite pains3 it  
is that indeed1 but it is also far more4
 ,genius means ideas1 intuition1 a      
faculty of seizing by thought the hidden
 relations of things1 and withal the    
power of proceeding step by step to     
their clear and full expression1 whether
 in the language of mathematical        
analysis or in the diction of daily     
life4 ,such was the genius of ,lord     
,kelvin2 it was lofty and it was        
practical4 ,he understood--for he had   
felt--the fascination of knowledge apart
 from its application to mechanical     
devices2 he did not disdain to devote   
his great powers to the service of      
mankind4 ,his objects of daily          
contemplation were the play of forces1  
the actions of bodies in all their      
varied manifestations1 or1 as he        
preferred to sum up the realm of        
physics1 the observation and discussion 
of properties of matter4 ,but his eyes  
were ever open to the bearing of all    
that he saw or discovered on the        

improvement of industrial appliances1 to
 the possibility of using it to increase
 the comfort and safety of men1 and so  
to augment the sum total of human       
happiness4  ,his statement1 which has   
been so often quoted1 that after        
fifty-five years of constant study he   
knew little more of electricity and     
magnetism than he did at the beginning  
of his career1 is not to be taken as a  
confession of failure4 ,it was1 like    
,newton's famous declaration1 an        
indication of his sense of the vastness 
of the ocean of truth and the           
manifoldness of the treasures which     
still lie within its deep unfathomed    
caves4 ,like ,newton1 he had merely     
wandered along the shore of that great  
ocean1 and here and there sounded its   
accessible depths1 while its infinite   
expanse lay unexplored4 ,and also like  
,newton--indeed like all great men--he  
stood with deep reverence before the    
great problems of the soul and destiny  
of man4 ,he believed that ,nature1 which

 he had sought all his life to know and 
understand1 showed everywhere the       
handiwork of an infinite and beneficent 
intelligence1 and he had faith that in  
the end all that appeared dark and      
perplexing would stand forth in fulness 
of light4     ,,footnotes3     #a ,lord 
,kelvin's address on his installation as
 ,chancellor of   the ,university of    
,glasgow1 ,november #bi1 #aijd4    #b   
,successor of ,dr4 ,dick1 the ,professor
 of ,natural ,philosophy   who induced  
the ,faculty to grant a workshop to     
,james ,watt when   the ,corporation of 
,hammermen prevented him from starting  
business in ,glasgow1 and for whom ,watt
 was repairing the   ,newcomen engine   
when he invented the separate condenser4
    #c ,a model steam-engine which he   
made in his youth was   carefully       
preserved by his brother in the ,natural
 ,philosophy   ,department4 ,it was     
homely but accurate in construction3 the
   beam was of wood1 and the piston was 
an old thick copper penny6    #d        

,proceedings on the occasion of the     
,presentation to the   ,university of   
,glasgow of the ,portrait of ,emeritus  
,professor ,g4   ,g4 ,ramsay4 ,november 
#f1 #aijg4    #e ,apparently for a short
 time in #ahda1 when ,dr4 ,meikleham was
   laid aside by illness4    #f ,the    
,c4,u4,m4,s4 began as a ,peterhouse     
society in #ahdc1 and   after a first   
concert1 which was followed by a supper1
 and that   by certain operations on the
 chapel roof1 the ,master would   only  
give permission to hold a second concert
 in the ,red ,lion   at ,cambridge1     
there being no room in ,college1 on     
condition that   the society called     
itself the ,university ,musical         
,society4 ,the   new society was formed 
in ,may #ahdd2 the first president was  
,g4   ,e4 ,smith1 of ,peterhouse1 the   
second was ,blow1 also of   ,peterhouse1
 a violin player and 'cellist1 and the  
third was   ,thomson4 ,see ,cambridge   
,chronicle1 ,july #aj1 #aijc1 and ,the  
,cambridge ,review1 ,feb4 #bj1 #aijh4   

#g ,it is rather strange that the ninth 
edition of the   ,encyclopdia           
,britannica contains no biography of    
,green4 ,born   in the year #agic at    
,nottingham1 the son of a baker1 he     
assisted   his father1 who latterly     
acquired a miller's business at the     
neighbouring village of ,sneinton4 ,in  
#ahbi his father died1 and   he disposed
 of the business in order that he might 
have leisure   to give to mathematics1  
in which1 though entirely self-taught1  
he had begun to make original           
researches4 ,his famous ',essay'   was  
published by subscription in #ahbh1 and 
attracted but little   attention4 ,in   
#ahcc1 at forty years of age1 ,green    
entered at   ,gonville and ,caius       
,college1 and obtained the fourth place 
in   the mathematical tripos of #ahcg1  
the year of ,griffin1   ,sylvester1 and 
,gregory4 ,his university career1       
whatever else it   may have done1       
apparently did not tend to make his     
earlier work   much better known to the 

general scientific public1 and he died  
in #ahda without the scientific         
recognition which was his due4   ,that  
came later when1 as stated below1       
,thomson discovered him   to the ,french
 mathematicians and republished his     
',essay4'    #h ,january #ahfi1         
,reprint1 etc41 ,article ,,xv4    #i    
,reprint1 ,article ,v4    #aj ,the      
geometrical idea was1 however1 given and
 applied at   least as early as #ahcf by
 ,bellavitis1 for a paper entitled      
,teoria delle figure inversa appears in 
the ,annali delle   ,scienze del ,regno 
,lombardo-,veneto for that year4 ,it was
 also   described as an independent     
discovery by ,mr4 ,john ,wm4 ,stubbs1   
in a paper in the ,philosophical        
,magazine for ,november #ahdc4   ,in a  
note on the history of the              
transformation in ,taylor's   ,geometry 
of ,conics the date 7without reference7 
of ,bellavitis   is given1 and it is    
stated that the method of inversion was 
given afresh by ,messrs4 ,ingram and    

,stubbs 7,dublin1 ,phil4 ,soc4   ,trans4
 ,i74 ,the note also mentions that      
inversion was applied   by ,dr4 ,hirst  
to attractions1 but contains no         
reference to   ,thomson's papers6    #aa
 ,de ,caloris distributione per ,terr   
,corpus in the   ,faculty minute1 as    
stated above4    #ab ,sic4 ,without     
doubt a mistake of the scribe for       
,liouville4    #ac ,north ,wales        
,chronicle1 ,report1 ,feb4 #g1 #ahhe4   
#ad ,published3 ,treatise on ,natural   
,philosophy1 vol4 i in   #ahfg2         
,elements of ,natural ,philosophy in    
#ahgc4    #ae ,the exact date at which  
this was done cannot be determined      
from the ,minutes of the ,faculty1 as   
they contain no reference   to the      
appropriation of space for the purpose4 
,in his ,oration   on ,james ,watt1     
delivered at the ,ninth ,jubilee of the 
,university of ,glasgow1 in #aija1 ,lord
 ,kelvin referred to the   ,glasgow     
,physical ,laboratory as having grown up
 between #ahdf and   #ahef2 and         

elsewhere he has referred to it as      
having been   incipient in #ahea4    #af
 ,there are now in ,glasgow in the      
winter session alone about   #cfj       
elementary students and #hj advanced    
students1 and about #bej   taking       
practical laboratory work4    #ag       
,before his death 7in #ahcb7 ,carnot had
 obtained a clear   perception of the   
true state of the case1 and of the      
complete   doctrine of the conservatism 
of energy4 ,see extracts from           
,carnot's unpublished writings appended1
 with a biography1 to   the reprinted   
,memoir1 by his younger brother1        
,hippolyte   ,carnot4    #ah ,this      
equation for the porous plug experiment 
may be   established in the following   
manner1 which forms a good example   of 
,thomson's second definition of absolute
 temperature4 ,take   pressure and      
volume of the gas on the supply side of 
the plug   as p  dp and v1 and on the   
delivery side as p and v  dv1 so   that 
dp and dv are positive4 ,the net work   

done in forcing the   gas through the   
plug  7p  dp7v  p7v  dv7   pdv  vdp4    
,let a heating effect result so that    
temperature is changed from   ,t to ,t  
,t4 ,let this be annulled by abstraction
 of heat   ,cp,t at constant pressure4  
7,cp  sp4 heat press4 const47   ,it is  
to be understood that dv is the total   
expansion   existing1 after this        
abstraction of heat4 ,the energy e of   
the   fluid has been increased by de    
pdv  vdp  ,cp,t4    ,now1 since the     
original temperature has been restored1 
the   same expansion dv if imposed      
isothermally would involve the   same   
energy change de2 but in that case heat 
d,h 7dynamical7   would be absorbed1 and
 work pdv would be done by the gas4     
,hence de  d,h  pdv4 ,this1 with the    
former value of de1 gives   d,h  vdp    
,cp,t4 ,thomson's work-ratio is thus    
pdv7vdp  ,cp,t74   ,now suppose dp      
imposed without change of volume1 and   
d,t to be the   resulting temperature   
change4 ,the temperature and pressure   

ratios   are d,t,t1 dpp4 ,thus d,t,t  dp
 dv7vdp  ,cp,t71 or      7v,t77d,tdv7   
#aa  7,cpv77,tdp7    which is ,thomson's
 equation4 ,the minus sign on the right 
arises   from a heating effect having   
been taken here as the normal   case4   
,if the temperature ,t is restored by   
removing the heat at   constant volume1 
a similar process gives the equation    
7v,t77d,tdv7  #a  7,tp77,tdp7#a         
7,cvv77,tdp7    where dp is the change  
of pressure before the restoration of   
the temperature ,t1 and ,tp is the rate 
of variation of ,t   with p1 volume     
constant4    #ai ,on a ,universal       
,tendency in ,nature to ,dissipation of 
,energy1 ,proc4 ,r4,s4,e41 #aheb1 and   
,phil4 ,mag41 ,oct41 #aheb4    #bj ,to  
this may be added the extremely useful  
theorem for such   problems1 that if any
 directed quantity ,l1 say1             
characteristic   of the motion of a     
body1 be associated with a line or axis 
,ol1   which is changing in direction1  
it causes a rate of production   of the 

same quantity for a line or axis        
instantaneously at   right angles to    
,ol1 towards which ,ol is turning with  
angular   velocity 1 of amount ,l4 ,if  
,m be the amount of the   quantity      
already existing for this latter line or
 axis1 the   total rate of growth of the
 quantity is there ,m  ,l4 ,for         
example1 a particle moving with uniform 
speed v in a circle of   radius r1 has  
momentum mv along the tangent4 ,but the 
tangent is   turning round as the       
particle moves with angular speed vr1   
towards the radius4 ,the rate of growth 
of momentum towards the   centre is     
therefore                  mv  vr  mvr4 
#ba ,see ,gray's ,lehrbuch der ,physik1 
s4 #bgh4 ,vieweg u4 ,sohn1   #aijd4     
#bb ,gray1 ,royal ,institution1 ,friday 
,evening ,discourse1   ,february #ahih4 
#bc ,see the ,reports of the ,committee 
on ,electrical ,standards1   edited by  
,prof4 ,fleeming ,jenkin1 ,f4,r4,s41    
,maxwell's ,electricity   and           
,magnetism1 and ,gray's ,theory and     

,practice of ,absolute   ,measurements  
in ,electricity and ,magnetism1 ,vol4   
,,ii1 ,part ,,ii4    #bd ,the writer    
once1 on a thick night1 in a passenger  
steamer   in the ,race of ,alderney1    
when the engines were stopped and       
soundings were being taken1 saw the reel
 and cord go overboard1   nearly taking 
one of the men with it4 ,a new hank of  
cord had to   be got and bent on a new  
reel2 an operation that took a long     
time1 during which the exact locality of
 the ship was a matter   of uncertainty4
 ,comment is needless6    #be ,the      
tuning of a major third1 in this way1 is
 described in   the paper entitled      
,beats on ,imperfect ,harmonies1        
published in   ,popular ,lectures and   
,addresses1 vol4 ii4    #bf ,the writer 
well remembers meeting a man of some    
experience   in cable work who was on   
his way to measure the alternating      
currents in a ,jablochkoff candle       
installation by the aid of an   ,ayrton 
and ,perry galvanometer with steel      

needle6     ,,index     ,atlantic       
cables1 #bfg1 #bfh    ,atmospheric      
electricity1 #bbf    ,atoms1 size of1   
#bfa    ,ayrton1 ,w4 ,e41 #bif          
,baltimore lectures1 #bed-bfc           
,bertrand's theorem of maximum kinetic  
energy1 #aeh    ,bottomley1 ,james      
,thomson1 #caa    ,bottomley1 ,william1 
#g    ,british ,association1 electrical 
standards1 #bdd-bec     ,cambridge      
,university ,musical ,society1 #bd      
,cambridge and ,dublin ,mathematical    
,journal1 #be1 #ca1 #gh    ,carnot1     
,sadi1 #gg1 #aja    ,carnot's ,thorie   
,motrice du ,feu1 #hg1 #aja1 #ajh et    
seq4    ,cauchy1 #bid    ,chasles1 #bh1 
#dc    ,clapeyron1 #aja1 #aab           
,clausius1 #aad et seq4    ,college1 the
 old1 of ,glasgow1 #aj    ,compass1     
errors of1 #bgc     ,dew-drop1          
artificial1 #bij    ,dynamical theorems1
 ,thomson's and ,bertrand's1 #aeh et    
seq4     ,earth1 the age of1 #aif1      
#bbi-bdc    ,earth1 tidal retardation   
of1 #bcj    ,elasticity1                

,poisson-,navier theory of1 #bia2       
encyclopdia article on1 #big            
,electrical oscillations1 #aha et seq4  
,electricity1 mathematical theory of1   
#cc    ,electrolysis1 mechanical theory 
of1 #agf    ,electrometers1 #bbc et seq4
    ,electromotive forces1 estimation   
of1 by heats of combination1 #agh       
,electromotive forces1 measurement of1  
#agi    ,electrostatics and ,magnetism1 
#bbb et seq4    ,ellis1 ,robert ,leslie1
 #bf    ,energy1 dissipation of1 #aci   
,faculty1 the1 of the ,university of    
,glasgow1 #d1 #fc-fg    ,faraday1 #fa   
,faure1 ,m41 #ha    ,fitz,gerald1 ,g4   
,f41 #cja1 #cje    ,fourier1 ,thorie    
,analytique de la ,chaleur1 #af et seq4 
,gauss1 #bh    ,gauss and ,weber1 #bde  
,green1 ,george1 of ,nottingham1 #ba1   
#cj1 #bid    ,gregory1 ,j4 ,w41 #bda    
,goodwin1 ,harvey1 #bf    ,gyrostats and
 gyrostatic action1 #bad1 #bhd-bhf      
,hamilton1 ,sir ,william ,rowan1 #aif1  
#bid    ,heat1 encyclopdia article on1  
#big    ,heaviside1 ,oliver1 #bid       

,helmholtz1 von1 #aac    ,hertz1 #aia1  
#bef    ,hopkins1 ,william1 #bc         
,huxley1 #gg1 #aif1 #bdb                
,hydrodynamics1 #aec-age     ,images1   
electric1 #ca1 #ch-ei    ,inversion1    
electrical1 #di et seq4    ,inversion1  
geometrical1 #ei1 #fj     ,joule1 ,james
 ,prescott1 #gg1 #hf et seq41 #aja et   
seq4     ,larmor1 ,joseph1 #bef         
,lectures on ,natural ,philosophy at    
,glasgow1 #bgi et seq4    ,liouville1   
#ca    ,liouville's ,journal de         
,mathmatiques1 #be1 #bf1 #ca            
,loschmidt1 #bfb    ,lubbock1 ,sir ,john
 7,lord ,avebury71 #he    ,luminiferous 
ether1 motion of planets through1 #bef  
,magnetism1 theory of1 #bbg             
,mariners' compass1 #bgb et seq4        
,maxwell1 #aag1 #aic1 #cje    ,mayer1 of
 ,heilbronn1 #aje    ,mc,farlane1       
,donald1 #if1 #bhg1 #bhi    ,mc,kichan1 
,dugald1 #aic    ,mcanique ,analytique  
of ,lagrange1 #aii1 #bje    ,mcanique   
,cleste of ,laplace1 #aii1 #bje         
,meikleham1 ,william1 #fa    ,mirror    

galvanometer1 #bfh    ,motivity1        
thermodynamic1 #ach     ,natural        
,philosophy1 ,chair of1 at ,glasgow1 #fc
    ,natural ,philosophy1 ,thomson and  
,tait's1 #aif et seq4    ,navigational  
sounding machine1 #bgb    ,newton1 #aie1
 #bjb    ,nichol1 ,john1 ,professor of  
,english ,language and ,literature1 #e  
,nichol1 ,john ,pringle1 ,professor of  
,astronomy1 #e1 #bj1 #fa1 #fc           
,oersted1 #fa    ,oscillations1         
electrical1 #aha et seq4     ,parkinson1
 ,stephen1 #bg    ,peltier1 #adh        
,pendulum1 ballistic1 #bhh    ,perry1   
,john1 #bdj1 #bif    ,phosphorescence1  
dynamical theory of1 #bei    ,physical  
laboratory1 first1 #gj    ,pickering1   
#bag    ,polarised light1 rotation of   
plane of1 #bbj    ,principia1 ,newton's1
 #aie1 #bjb     ,ramsay1 ,george        
,gilbert1 ,professor of ,humanity1 #aa  
,regnault1 #bi    ,royal ,society of    
,edinburgh1 presidency of1 #bii         
,royal ,society of ,london1 presidency  
of1 #bii    ,rumford1 ,count1 #ajc      

,seebeck1 #adh    ,signalling1 theory of
 telegraphic1 #bfd    ,siphon recorder1 
#bfh1 #bgj    ,smith1 ,archibald1 #bge  
,spectrum analysis1 dynamical theory of1
 #hd    ,stokes1 ,sir ,george ,gabriel1 
#bd1 #gi1 #hj1 #ha1 #he1 #bia1 #bid     
,stoney1 ,dr4 ,johnstone1 #bfb    ,sun's
 heat1 age of1 #bcb     ,tait1 ,peter   
,guthrie1 #aid et seq4    ,temperature1 
absolute1 #abe et seq42     comparison  
of1 with scale of air thermometer1 #ace 
,thermodynamics1 #ii-aeb                
,thermoelasticity1 #adb et seq4         
,thermoelectricity1 #adg et seq4        
,thermometry1 absolute1 #aad-aeb        
,thomson1 ,david1 #fa    ,thomson1      
,james1 ,professor of ,mathematics1     
#a-d1 #g    ,thomson1 ,james1 ,professor
 of ,engineering1 #aac1 #bji2           
integrating machine1 #bji1 #cjc         
,thomson and ,tait's ,natural           
,philosophy1 #fh1 #aif et seq41 #bah    
,thomson's theorem of minimum kinetic   
energy1 #aeh    ,thomson1 ,thomas1      
,professor of ,chemistry1 #f            

,thomson1 prevalence of name at ,glasgow
 ,college1 #e    ,thomson1 ,william1    
,lord ,kelvin3--     ,parentage and     
early education1 #a-ab     ,career at   
,universities of ,glasgow and           
,cambridge1 #ac-cb     ,early           
researches1 #af1 #ah1 #ca     ,election 
to ,chair of ,natural ,philosophy at    
,glasgow1 #fd     ,scientific           
researches1 passim2       ,jubilee of1  
#cja2       ,chancellor of ,university  
of ,glasgow1 #cjb     ,in class-room and
 laboratory1 #bgi-bih     ,practical    
activities1 honours and distinctions1   
last illness and         death1         
#bii-cjd2       funeral in ,westminster 
,abbey1 #cjd    ,tidal ,analyser1 #baa  
,tide ,predicter1 #bjh                  
,vortex-,motion1 #afa-age     ,waldstein
 sonata1 #bd    ,weber1 ,w41 #aic       
,weights and measures1 ,british1 #bhi1  
#bij    ,white1 ,james1 #bgf    ,willard
 ,gibbs1 #bid      ,,richard ,,clay     
,,sons1 ,,limited1   ,,bread ,,street   
,,hill1 ,e4,c41 ,,and   ,,bungay1       

,,suffolk4