,on a ,dynamical ,top1      = exhibit+ !
 ph5om5a ( ! motion ( a sy/em (         
inv>iable  =m ab\t a fix$ po9t1 ) "s    
sugge/ions z to ! ,e>?'s  motion  ,james
 ,cl]k ,maxwell    ,f ! ,transactions ( 
! ,royal ,society ( ,ed9burgh1 ,vol4    
,,xxi4  ,"p ,,iv4  7,read #bj? ,april1  
#aheg47      ,to ^? :o /udy ! progress (
 exact sci5ce1 ! common  sp9n+-top is a 
symbol ( ! lab\rs & ! p]plexities ( m5  
:o  _h successfully ?read$ ! mazes ( !  
planet>y motions4 ,!  ma!maticians ( !  
la/ age1 se>*+ ?r\gh nature = problems  
wor?y ( _! analysis1 f\nd 9 ?is toy ( _!
 y\?1 ample  occupation = _! highe/     
ma!matical p[]s4    ,no illu/ration (   
a/ronomical precession c be devis$ m    
p]fect  ?an t pres5t$ by a prop]ly      
balanc$ top1 b yet ! motion (  rotation 
has intricacies f> exce$+ ^? ( ! !ory ( 
precession4    ,accord+ly1 we f9d ,eul] 
& ,d',alemb]t devot+ _! tal5t &  _!     
pati5ce to ! e/abli%m5t ( ! laws ( !    
rotation (  solid bodies4 ,lagrange has 
incorporat$ 8 own analysis ( ! problem  

) 8 g5]al treatm5t ( me*anics1 & s9ce 8 
"t ,m4 ,po9st  has br\ght ! subject "u !
 p[] ( a m se>*+ analysis  ?an t ( !    
calculus1 9 :i* ideas take ! place (    
symbols1  & intelligible propositions   
sup]s$e equations4    ,9 ! practical    
dep>tm5t ( ! subject1 we mu/ notice !   
rotatory  ma*9e ( ,bohn5b]g]1 & !       
nautical top ( ,tr\ghton4 ,9 !  fir/ (  
^! in/rum5ts we h ! model ( !           
,gyroscope1 by :i*  ,f\cault has be5    
able to r5d] visible ! effects ( ! e>?'s
  rotation4 ,! beautiful exp]im5ts by   
:i* ,mr ,j4 ,elliot has made !  ideas ( 
precession s famili> to u are p]=m$ ) a 
top1 simil>  9 "s respects to           
,tr\ghton's1 ?\gh n borr[$ f 84    ,!   
top :i* ,i h ! hon\r to sp9 be=e !      
,society1 diff]s  f t ( ,mr ,elliot 9   
hav+ m adju/m5ts1 & 9 be+  design$ to   
exhibit f> m complicat$ ph5om5a4    ,!  
arrangem5t ( ^! adju/m5ts1 s z to       
produce ! desir$  effects1 dep5ds on !  
ma!matical !ory ( rotation4 ,! me?od (  
exhibit+ ! motion ( ! axis ( rotation1  

by means ( a col\r$  disc1 is ess5tial  
to ! success ( ^! adju/m5ts4 ,?is       
optical  contrivance = r5d]+ visible !  
nature ( ! rapid motion ( !  top1 & !   
practical me?ods ( apply+ ! !ory (      
rotation to  su* an in/rum5t z ! "o be=e
 u1 are ! gr\nds on :i* ,i  br+ my      
in/rum5t & exp]im5ts be=e ! ,society z  
my own4    ,i propose1 !re=e1 9 ! fir/  
place1 to give a brief outl9e (  su*    
p>ts ( ! !ory ( rotation z are necess>y 
= !  explanation ( ! ph5om5a ( ! top4   
,i %all !n describe ! in/rum5t ) its    
adju/m5ts1 & !  effect ( ea*1 ! mode (  
obs]v+ ( ! col\r$ disc :5 ! top  is 9   
motion1 & ! use ( ! top 9 illu/rat+ !   
ma!matical  !ory1 ) ! me?od ( mak+ !    
diff]5t exp]im5ts4    ,la/ly1 ,i %all   
attempt to expla9 ! nature ( a possible 
v>iation  9 ! e>?'s axis due to its     
figure4 ,?is v>iation1 if x exi/s1  mu/ 
cause a p]iodic inequality 9 ! latitude 
( e place on !  e>?'s surface1 go+ ?r\gh
 its p]iod 9 ab\t elev5 mon?s4 ,!  am\nt
 ( v>iation mu/ be v small1 b its "*    

gives x  importance1 & ! necess>y       
obs]vations are already made1 & only    
require r$uction4    ,on ! ,!ory (      
,rotation4      ,! !ory ( ! rotation ( a
 rigid sy/em is /rictly d$uc$ f  !      
elem5t>y laws ( motion1 b ! complexity (
 ! motion ( !  p>ticles ( a body freely 
rotat+ r5d]s ! subject s intricate1  t x
 has nev] be5 ?or\ghly und]/ood by any b
 ! mo/ exp]t  ma!maticians4 ,_m :o h    
ma/]$ ! lun> !ory h come to  errone\s   
conclusions on ?is subject2 & ev5       
,newton has *os5 to  d$uce ! di/urbance 
( ! e>?'s axis f 8 !ory ( !  motion ( ! 
nodes ( a free orbit1 r ?an attack !    
problem (  ! rotation ( a solid body4   
,! me?od by :i* ,m4 ,po9st has r5d]$ !  
!ory m manageable1  is by ! lib]al      
introduction ( 8appropriate ideas10     
*iefly ( a  geometrical "*1 mo/ ( :i* _h
 be5 r5d]$ famili> to  ma!maticians by !
 writ+s ( ,monge1 b :i* !n fir/ became  
illu/rations ( ?is bran* ( dynamics4 ,if
 any fur!r progress is to  be made 9    
simplify+ & arrang+ ! !ory1 x mu/ be by 

!  me?od :i* ,po9st has repeat$ly po9t$ 
out z ! only "o :i*  c lead to a true k 
( ! subject1--t ( proce$+ f  "o di/9ct  
idea to ano!r in/ead ( tru/+ to symbols 
&  equations4    ,an important          
contribution to our /ock ( appropriate  
ideas & me?ods  has lately be5 made by  
,mr ,r4 ,b4 ,hayw>d1 9 a pap]1 8,on a   
,direct  ,me?od ( e/imat+ ,velocities1  
,accel]ations1 & all simil>  quantities1
 ) respect to axes1 moveable 9 any mann]
 9 ,space40  7,trans4 ,cambridge ,phil4 
,soc ,vol4 x4 ,"p ,i47    99 ,9 ?is     
communication ,i int5d to conf9e myself 
to t "p ( !  subject :i* ! top is int5d$
 io illu/rate1 namely1 ! alt]ation  ( ! 
position ( ! axis 9 a body rotat+ freely
 ab\t its c5tre  ( gravity4 ,i %all1    
!re=e1 d$uce ! !ory z briefly z         
possible1 f two consid]ations only1--!  
p]man5ce ( ! orig9al  angul> mom5tum 9  
direction & magnitude1 & ! p]man5ce (  !
 orig9al vis viva4    99 ,! ma!matical  
difficulties ( ! !ory ( rotation arise  
*iefly  f ! want ( geometrical          

illu/rations & s5sible images1 by  :i*  
we might fix ! results ( analysis 9 our 
m9ds4    ,x is easy to und]/& ! motion (
 a body revolv+ ab\t a fix$  axle4 ,e   
po9t 9 ! body describes a circle ab\t ! 
axis1 &  returns to its orig9al position
 aft] ea* complete revolution4 ,b if  ! 
axle itself be 9 motion1 ! pa?s ( !     
diff]5t po9ts ( !  body w no long] be   
circul> or re-entrant4 ,ev5 ! velocity (
  rotation ab\t ! axis requires a c>eful
 def9ition1 & !  proposition t1 9 all   
motion ab\t a fix$ po9t1 !re is always  
"o l9e ( p>ticles =m+ an in/antane\s    
axis1 is usually giv5  9 ! =m ( a v     
repulsive mass ( calculation4 ,mo/ ( ^! 
difficulties may be got rid ( by devot+ 
a little att5tion to !  me*anics &      
geometry ( ! problem be=e ent]+ on !    
discussion  ( ! equations4    ,mr       
,hayw>d1 9 8 pap] already ref]r$ to1 has
 made great use ( !  me*anical          
conception ( ,angul> ,mom5tum4          
,def9ition #a ,! ,angul> ,mom5tum ( a   
p>ticle ab\t an axis is  measur$ by !   

product ( ! mass ( ! p>ticle1 its       
velocity  resolv$ 9 ! normal plane1 & ! 
p]p5dicul> f ! axis on  ! direction (   
motion4    99 ,! angul> mom5tum ( any   
sy/em ab\t an axis is ! algebraical  sum
 ( ! angul> mom5ta ( its p>ts4    ,z !  
rate ( *ange ( ! l9e> mom5tum ( a       
p>ticle measures  ! mov+ =ce :i* acts on
 x1 s ! rate ( *ange (  angul> mom5tum  
measures ! mom5t ( t =ce ab\t an axis4  
,all actions betwe5 ! p>ts ( a sy/em1   
be+ pairs ( equal &  opposite =ces1     
produce equal & opposite *anges 9 !     
angul>  mom5tum ( ^? p>ts4 ,h5ce ! :ole 
angul> mom5tum ( ! sy/em  is n affect$  
by ^! actions & re-actions4    99 ,:5 a 
sy/em ( inv>iable =m revolves ab\t an   
axis1 ! angul>  velocity ( e "p is !    
same1 & ! angul> mom5tum ab\t !  axis is
 ! product ( ! angul> velocity & ! mom5t
 (  in]tia ab\t t axis4    99 ,x is only
 9 p>ticul> cases1 h[ev]1 t ! :ole      
angul>  mom5tum c be e/imat$ 9 ?is way4 
,9 g5]al1 ! axis ( angul>  mom5tum      
diff]s f ! axis ( rotation1 s t !re w be

 a  residual angul> mom5tum ab\t an axis
 p]p5dicul> to t (  rotation1 unless t  
axis has "o ( ?ree positions1 call$ !   
pr9cipal axes ( ! body4    ,by ref]r+   
ev]y?+ to ^! ?ree axes1 ! !ory is       
greatly  simplifi$4 ,! mom5t ( in]tia   
ab\t "o ( ^! axes is great]  ?an t ab\t 
any o!r axis ?r\gh ! same po9t1 & t ab\t
  "o ( ! o!rs is a m9imum4 ,^! two are  
at "r angles1 & !  ?ird axis is         
p]p5dicul> to _! plane1 & is call$ !    
mean  axis4    99 ,let ,a1 ,b1 ,c be !  
mom5ts ( in]tia ab\t ! pr9cipal axes    
?r\gh ! c5tre ( gravity1 tak5 9 ord] (  
magnitude1 & let  omega#a omega#b       
omega#c be ! angul> velocities ab\t !m1 
!n ! angul> mom5ta w be ,aomega#a1      
,bomega#b1 &  ,comega#c4    ,angul>     
mom5ta may be comp\nd$ l =ces or        
velocities1 by ! law  ( !               
8p>allelogram10 & s9ce ^! ?ree are at "r
 angles to  ea* o!r1 _! resultant is    
beg9displayma? sqrt,a#bomega#ab         
,b#bomega#bb   ,c#bomega#cb  ,h         
enddisplayma? 7#a7    & ?is mu/ be      

con/ant1 bo? 9 magnitude & direction 9  
space1  s9ce no ext]nal =ces act on !   
body4    ,we %all call ?is axis ( angul>
 mom5tum ! inv>iable axis4 ,x  is       
p]p5dicul> to :at has be5 call$ !       
inv>iable plane4 ,po9st  calls x ! axis 
( ! c\ple ( impulsion4 ,!               
direction-cos9es  ( ?is axis 9 ! body   
are1      beg9displayma? beg9arrayc c c 
display/yle l   frac,aomega#a;,h1 444   
444ga#b;,h1  display/yle n              
frac,comega#c;,h4 endarrayenddisplayma? 
,s9ce ,i1 m & n v>y dur+ ! motion1 we   
ne$ "s additional  condition to det]m9e 
! relation betwe5 !m4 ,we f9d ?is 9 !   
prop]ty ( ! vis viva ( a sy/em (        
inv>iable =m 9 :i*  !re is no friction4 
,! vis viva ( su* a sy/em mu/ be        
con/ant4  ,we express ?is 9 ! equation  
beg9displayma? ,aomega#ab  ,bomega#bb   
,comega#cb  ,v  enddisplayma? 7#b7      
,sub/itut+ ! values ( omega#a1 omega#b1 
omega#c 9 t]ms  ( l1 m1 n1              
beg9displayma? fracl#b;,a  fracm#b;,b   
fracn#b;,c   frac,v,h#b4 enddisplayma?  

,let #a_/;,a  a#b1 #a_/;,b  b#b1 #a_/;c 
c#b1 ,v_/,h#b  e#b1 & ?is  equation     
becomes      beg9displayma? a#bl#b      
b#bm#b  c#bn#b  e#b  enddisplayma? 7#c7 
& ! equation to ! cone1 describ$ by !   
inv>iable axis )9  ! body1 is           
beg9displayma? 7a#b - e#b7 x#b  7b#b -  
e#b7 y#b  7c#b - e#b7 z#b   #j          
enddisplayma? 7#d7      ,! int]sections 
( ?is cone ) planes p]p5dicul> to !     
pr9cipal axes are f\nd by putt+ x1 y1 or
 z1 con/ant 9 ?is  equation4 ,by giv+ e 
v>i\s values1 all ! diff]5t pa?s ( !    
pole ( ! inv>iable axis1 correspond+ to 
diff]5t initial  circum/ances1 may be   
trac$4      ,figure3 ,figure #a      99 
,9 ! figures1 ,i h suppos$ a#b  #ajj1   
b#b #ajg1 & c#b  #aaj4 ,! fir/ figure   
repres5ts a section ( ! v>i\s cones by a
  plane p]p5dicul> to ! axis ( x1 :i* is
 t ( greate/  mom5t ( in]tia4 ,^!       
sections are ellipses hav+ _! major axis
  p>allel to ! axis ( b4 ,! value ( e#b 
correspond+ to ea*  ( ^! curves is      
indicat$ by figures beside ! curve4 ,!  

ellipticity increases ) ! size ( !      
ellipse1 s t ! section  correspond+ to  
e#b  #ajg w\ld be two p>allel /raight   
l9es  7beyond ! b\nds ( ! figure71 aft] 
:i* ! sections w\ld be  hyp]bolas4      
,figure3 ,figure #b      99 ,! second   
figure repres5ts ! sections made by a   
plane1  p]p5dicul> to ! mean axis4 ,!y  
are all hyp]bolas1 except :5  e#b  #ajg1
 :5 ! section is two int]sect+ /raight  
l9es4      ,figure3 ,figure #c      ,!  
?ird figure %[s ! sections p]p5dicul> to
 ! axis ( lea/  mom5t ( in]tia4 ,f e#b  
#aaj to e#b  #ajg ! sections are        
ellipses1 e#b  #ajg gives two p>allel   
/raight l9es1 & beyond  ^! ! curves are 
hyp]bolas4      ,figure3 ,figure #d     
99 ,! f\r? & fif? figures %[ ! sections 
( ! s]ies ( cones  made by a cube & a   
sph]e respectively4 ,! use ( ^! figures 
is  to exhibit ! connexion betwe5 !     
diff]5t curves describ$ ab\t  ! ?ree    
pr9cipal axes by ! inv>iable axis dur+ !
 motion (  ! body4      ,figure3 ,figure
 #e      99 ,we h next to comp>e !      

velocity ( ! inv>iable axis )  respect  
to ! body1 ) t ( ! body itself r\nd "o (
 !  pr9cipal axes4 ,s9ce ! inv>iable    
axis is fix$ 9 space1 its motion        
relative to ! body mu/ be equal &       
opposite to t ( ! portion  ( ! body     
?r\gh :i* x passes4 ,n[ ! angul>        
velocity ( a  portion ( ! body ^:       
direction-cos9es are l1 m1 n1 ab\t  !   
axis ( x is      beg9displayma?         
fracomega#aa - l#b - fracl#a -          
l#b7lomega#a  momega#b  nomega-#c74     
enddisplayma?      ,sub/itut+ ! values (
 omega#a1 omega#b1 omega#c1 9 t]ms  ( l1
 m1 n1 & tak+ acc\nt ( equation 7#c71   
?is expression  becomes                 
beg9displayma? ,hfrac7a#b - e#b7a -     
l#bl4 enddisplayma?      ,*ang+ ! sign &
 putt+ display/yle l  fracomega#a;a#b;,h
  we h ! angul> velocity ( ! inv>iable  
axis ab\t t ( x      beg9displayma?     
fracomega#aa - l#b frace#b - a#b;a#b1   
enddisplayma?      always positive ab\t 
! axis ( greate/ mom5t1 negative ab\t t 
( lea/ mom5t1 & positive or negative    

ab\t ! mean axis accord+  to ! value (  
e#b4 ,! direction ( ! motion 9 e case is
  repres5t$ by ! arr[s 9 ! figures4 ,!  
arr[s on ! outside (  ea* figure        
indicate ! direction ( rotation ( !     
body4    99 ,if we att5d to ! curve     
describ$ by ! pole ( ! inv>iable  axis  
on ! sph]e 9 fig4 #e1 we %all see t !   
areas describ$ by  t po9t1 if project$  
on ! plane ( yz1 are swept out at !     
rate      beg9displayma? omega#a frace#b
 - a#b;a#b4 enddisplayma?      ,n[ !    
semi-axes ( ! projection ( ! sph]ical   
ellipse describ$  by ! pole are         
beg9displayma? sqrtfrace#b - a#b;b#b -  
a#b  hspace#a;cmtextrm&hspace#a;cm      
sqrtfrace#b - a#b;c#b -  a#b4           
enddisplayma?      ,divid+ ! area ( ?is 
ellipse by ! area describ$ dur+ "o      
revolution ( ! body1 we f9d ! numb] (   
revolutions ( ! body  dur+ ! description
 ( ! ellipse--      beg9displayma?      
fraca#bsqrtb#b - a#bsqrtc#b - a#b4      
enddisplayma?      ,! projections ( !   
sph]ical ellipses ^u ! plane ( yz are   

all simil> ellipses1 & describ$ 9 ! same
 numb] ( revolutions2  & 9 ea* ellipse s
 project$1 ! area describ$ 9 any "t is  
proportional to ! numb] ( revolutions ( 
! body ab\t ! axis (  x1 s t if we      
measure "t by revolutions ( ! body1 !   
motion  ( ! projection ( ! pole ( !     
inv>iable axis is id5tical )  t ( a body
 act$ on by an attractive c5tral =ce    
v>y+ directly  z ! di/ance4 ,9 ! case ( 
! hyp]bolas 9 ! plane ( !  greate/ &    
lea/ axis1 ?is =ce mu/ be suppos$       
repulsive4 ,!  dots 9 ! figures #a1 #b1 
#c1 are int5d$ to indicate r\ghly !     
progress made by ! inv>iable axis dur+  
ea* revolution ( ! body  ab\t ! axis (  
x1 y & z respectively4 ,x mu/ be        
rememb]$  t ! rotation ab\t ^! axes     
v>ies ) _! incl9ation to !  inv>iable   
axis1 s t ! angul> velocity dim9i%es z !
  incl9ation increases1 & !re=e ! areas 
9 ! ellipses above  m5tion$ are n       
describ$ ) uni=m velocity 9 absolute "t1
 b  are less rapidly swept out at !     
extremities ( ! major axis ?an at  ^? ( 

! m9or4    99 ,:5 two ( ! axes h equal  
mom5ts ( in]tia1 or b  c1 !n  ! angul>  
velocity omega#a is con/ant1 & ! pa? ( !
  inv>iable axis is circul>1 ! numb] (  
revolutions ( ! body  dur+ "o circuit ( 
! inv>iable axis1 be+                   
beg9displayma? fraca#b;b#b - a#b        
enddisplayma?      ,! motion is 9 ! same
 direction z t ( ! rotation1 or 9 !     
opposite direction1 accord+ z ! axis ( x
 is t ( greate/ or  ( lea/ mom5t (      
in]tia4    99 ,bo? 9 ?is case1 & 9 t 9  
:i* ! ?ree axes are unequal1  ! motion (
 ! inv>iable axis 9 ! body may be r5d]$ 
v sl[  by dim9i%+ ! diff]5ce ( ! mom5ts 
( in]tia4 ,! angul>  velocity ( ! axis (
 x ab\t ! inv>iable axis 9 space is     
beg9displayma? omega#a;frace#b -        
a#bl#b;a#b7a - l#b71  enddisplayma?     
:i* is great] or less ?an omega#a1 z e#b
 is great] or less  ?an a#b1 &1 :5 ^!   
quantities are ne>ly equal1 is v ne>ly  
! same z omega#a itself4 ,?is quantity  
indicates ! rate (  revolution ( ! axle 
( ! top ab\t its mean position1 & is v  

easily obs]v$4    99 ,! in/antane\s axis
 is n s easily obs]v$4 ,x revolves r\nd 
! inv>iable axis 9 ! same "t ) ! axis ( 
x1 at a  di/ance :i* is v small 9 ! case
 :5 a1 b1 c1 are ne>ly  equal4 ,f its   
rapid angul> motion 9 space1 & its ne>  
co9cid5ce  ) ! inv>iable axis1 !re is no
 advantage 9 /udy+ its motion  9 ! top4 
99 ,by mak+ ! mom5ts ( in]tia v unequal1
 & 9 def9ite  proportion to ea* o!r1 &  
by draw+ a few /rong l9es z  diamet]s ( 
! disc1 ! comb9ation ( motions w produce
 an  appe>ance ( epicycloids1 :i* are ! 
result ( ! cont9u$  int]section ( !     
successive positions ( ^! l9es1 & !     
cusps  ( ! epicycloids lie 9 ! curve 9  
:i* ! in/antane\s axis  travels4 ,"s ( !
 figures produc$ 9 ?is way are v pleas+4
    ,9 ord] to illu/rate ! !ory (       
rotation exp]im5tally1 we mu/  h a body 
balanc$ on its c5tre ( gravity1 &       
capable ( hav+  its pr9cipal axes &     
mom5ts ( in]tia alt]$ 9 =m & position   
)9 c]ta9 limits4 ,we mu/ be able to make
 ! axle ( !  in/rum5t ! greate/1 lea/1  

or mean pr9cipal axis1 or to make x  n a
 pr9cipal axis at all1 & we mu/ be able 
to see ! position  ( ! inv>iable axis ( 
rotation at any "t4 ,!re mu/ be ?ree    
adju/m5ts to regulate ! position ( !    
c5tre ( gravity1 ?ree  = ! magnitudes ( 
! mom5ts ( in]tia1 & ?ree = !           
directions ( ! pr9cipal axes1 n9e       
indep5d5t adju/m5ts1 :i*  may be        
di/ribut$ z we please among ! screws ( !
 in/rum5t4      ,figure3 ,figure #f     
,! =m ( ! body ( ! in/rum5t :i* ,i h    
f\nd mo/ suitable  is t ( a bell 7fig4  
#f74 ,c is a holl[ cone ( brass1 ,r is a
  heavy r+ ca/ 9 ! same piece4 ,six     
screws1 ) heavy heads1 x1  y1 z1 x'1 y'1
 z'1 "w horizontally 9 ! r+1 & ?ree     
simil> screws1 l1 m1 n1 "w v]tically    
?r\gh ! r+ at  equal int]vals4 ,,z is ! 
axle ( ! in/rum5t1 ,,ss is a brass      
screw work+ 9 ! upp] "p ( ! cone ,c1 &  
capable ( be+  firmly clamp$ by means ( 
! nut c4 ,b is a cyl9drical brass bob1  
:i* may be screw$ up or d[n ! axis1 &   
fix$ 9 its place by !  nut b4    ,! l[] 

extremity ( ! axle is a f9e /eel po9t1  
f9i%$ )\t  em]y1 & aft]w>ds h>d5$4 ,x   
runs 9 a little agate cup set 9  ! top (
 ! pill> ,p4 ,if any em]y _h be5 emb$d$ 
9 ! /eel1  ! cup w\ld soon be worn out4 
,! upp] end ( ! axle has also a  /eel   
po9t by :i* x may be kept /eady :ile    
sp9n+4    ,:5 ! in/rum5t is 9 use1 a    
col\r$ disc is atta*$ to ! upp]  end ( !
 axle4    ,x w be se5 t !re are elev5   
adju/m5ts1 n9e screws 9 !  brass r+1 !  
axle screw+ 9 ! cone1 & ! bob screw+ on 
!  axle4 ,! advantage ( ! la/ two       
adju/m5ts is1 t by !m l>ge  alt]ations c
 be made1 :i* are n possible by means ( 
! small  screws4    ,! fir/ ?+ to be    
done ) ! in/rum5t is1 to make ! /eel    
po9t at ! end ( ! axle co9cide ) ! c5tre
 ( gravity ( !  :ole4 ,?is is done      
r\ghly by screw+ ! axle to ! "r place   
ne>ly1 & !n balanc+ ! in/rum5t on its   
po9t1 & screw+  ! bob & ! horizontal    
screws till ! in/rum5t w rema9  balanc$ 
9 any position 9 :i* x is plac$4    ,:5 
?is adju/m5t is c>efully made1 !        

rotation ( ! top has no  t5d5cy to %ake 
! /eel po9t 9 ! agate cup1 h[ev]        
irregul>  ! motion may appe> to be4     
,! next ?+ to be done1 is to make "o ( !
 pr9cipal axes ( !  c5tral ellipsoid    
co9cide ) ! axle ( ! top4    ,to effect 
?is1 we mu/ beg9 by sp9n+ ! top g5tly   
ab\t its  axle1 /eady+ ! upp] "p ) ! f+]
 at fir/4 ,if ! axle is  already a      
pr9cipal axis ! top w cont9ue to revolve
 ab\t its  axle :5 ! f+] is remov$4 ,if 
x is n1 we obs]ve t ! top  beg9s to sp9 
ab\t "s o!r axis1 & ! axle moves away f 
!  c5tre ( motion & !n back to x aga91 &
 s on1 alt]nately  wid5+ its circles &  
contract+ !m4    ,x is impossible to    
obs]ve ?is motion successfully1 )\t !   
aid  ( ! col\r$ disc plac$ ne> ! upp]   
end ( ! axis4 ,?is disc  is divid$ into 
sectors1 & /rongly col\r$1 s t ea*      
sector may  be recognis$ by its col\r :5
 9 rapid motion4 ,if ! axis ab\t  :i* ! 
top is really revolv+1 falls )9 ?is     
disc1 its position  may be asc]ta9$ by !
 col\r ( ! spot at ! c5tre ( motion4    

,if ! c5tral spot appe>s r$1 we "k t !  
inv>iable axis at  t in/ant passes ?r\gh
 ! r$ "p ( ! disc4    ,9 ?is way we c   
trace ! motion ( ! inv>iable axis 9 !   
revolv+ body1 & we f9d t ! pa? :i* x    
describes ^u !  disc may be a circle1 an
 ellipse1 an hyp]bola1 or a /raight l9e1
  accord+ to ! arrangem5t ( ! in/rum5t4 
,9 ! case 9 :i* ! inv>iable axis        
co9cides at fir/ ) !  axle ( ! top1 &   
returns to x aft] sep>at+ f x = a "t1   
its true pa? is a circle or an ellipse  
hav+ ! axle 9 its  circumf]5ce4 ,! true 
pr9cipal axis is at ! c5tre ( ! clos$   
curve4 ,x mu/ be made to co9cide ) !    
axle by adju/+ !  v]tical screws l1 m1  
n4    ,suppose t ! col\r ( ! c5tre (    
motion1 :5 f>!/ f !  axle1 indicat$ t ! 
axis ( rotation pass$ ?r\gh ! sector    
,l1 !n ! pr9cipal axis mu/ also lie 9 t 
sector at half !  di/ance f ! axle4     
,if ?is pr9cipal axis be t ( greate/    
mom5t ( in]tia1 we mu/  raise ! screw l 
9 ord] to br+ x ne>] ! axle ,a4 ,if x   
be ! axis ( lea/ mom5t we mu/ l[] !     

screw l4 ,9 ?is way  we may make !      
pr9cipal axis co9cide ) ! axle4 ,let u  
suppose  t ! pr9cipal axis is t (       
greate/ mom5t ( in]tia1 & t  we h made x
 co9cide ) ! axle ( ! in/rum5t4 ,let u  
also  suppose t ! mom5ts ( in]tia ab\t !
 o!r axes are equal1 &  v little less   
?an t ab\t ! axle4 ,let ! top be spun   
ab\t  ! axle & !n receive a di/urbance  
:i* causes x to sp9 ab\t  "s o!r axis4  
,! in/antane\s axis w n rema9 at re/    
ei!r  9 space or 9 ! body4 ,9 space x w 
describe a "r cone1  complet+ a         
revolution 9 some:at less ?an ! "t (    
revolution (  ! top4 ,9 ! body x w      
describe ano!r cone ( l>g] angle 9 a    
p]iod :i* is long] z ! diff]5ce ( axes (
 ! body is  small]4 ,! inv>iable axis w 
be fix$ 9 space1 & describe a  cone 9 ! 
body4    ,! relation ( ! diff]5t motions
 may be und]/ood f !  foll[+            
illu/ration4 ,take a hoop & make x      
revolve ab\t a /ick  :i* rema9s at re/ &
 t\*es ! inside ( ! hoop4 ,! section  ( 
! /ick repres5ts ! pa? ( ! in/antane\s  

axis 9 space1  ! hoop t ( ! same axis 9 
! body1 & ! axis ( ! /ick  ! inv>iable  
axis4 ,! po9t ( contact repres5ts ! pole
 ( !  in/antane\s axis itself1 travell+ 
_m times r\nd ! /ick be=e  x gets once  
r\nd ! hoop4 ,x is easy to see t !      
direction 9  :i* ! hoop moves r\nd !    
/ick1 s t if ! top be sp9n+ 9  !        
direction ,l1 ,m1 ,n1 ! col\rs w appe> 9
 ! same ord]4    ,by screw+ ! bob ,b up 
! axle1 ! diff]5ce ( ! axes (  in]tia   
may be dim9i%$1 & ! "t ( a complete     
revolution ( !  inv>iable axis 9 ! body 
increas$4 ,by obs]v+ ! numb] (          
revolutions ( ! top 9 a complete cycle (
 col\rs ( ! inv>iable  axis1 we may     
det]m9e ! ratio ( ! mom5ts ( in]tia4    
,by screw+ ! bob up f>!r1 we may make ! 
axle ! pr9cipal axis  ( lea/ mom5t (    
in]tia4    ,! motion ( ! in/antane\s    
axis w !n be t ( ! po9t (  contact ( !  
/ick ) ! outside ( ! hoop roll+ on x4 ,!
  ord] ( col\rs w be ,n1 ,m1 ,l1 if !   
top be sp9n+ 9 !  direction ,l1 ,m1 ,n1 
& ! m ! bob is screw$ up1 ! m  rapidly w

 ! col\rs *ange1 till x ceases to be    
possible to make  ! obs]vations         
correctly4    ,9 calculat+ ! dim5sions (
 ! p>ts ( ! in/rum5t1 x is  necess>y to 
provide = ! exhibition ( ! in/rum5t )   
its axle  ei!r ! greate/ or ! lea/ axis 
( in]tia4 ,! dim5sions &  weights ( !   
p>ts ( ! top :i* ,i h f\nd mo/ suitable1
 are  giv5 9 a note at ! end ( ?is pap]4
    ,n[ let u make ! axes ( in]tia 9 !  
plane ( ! r+ unequal4  ,we may d ?is by 
screw+ ! balance screws x & x#a f>!r  f 
! axle )\t alt]+ ! c5tre ( gravity4     
,let u suppose ! bob ,b screw$ up s z to
 make ! axle ! axis  ( lea/ in]tia4 ,!n 
! mean axis is p>allel to xx#a1 & !     
greate/ is at "r angles to xx#a 9 !     
horizontal plane4 ,! pa?  ( ! inv>iable 
axis on ! disc is no long] a circle b an
  ellipse1 conc5tric ) ! disc1 & hav+   
its major axis p>allel  to ! mean axis  
xx#a4    ,! small] ! diff]5ce betwe5 !  
mom5t ( in]tia ab\t ! axle  & ab\t !    
mean axis1 ! m ecc5tric ! ellipse w be2 
&  if1 by screw+ ! bob d[n1 ! axle be   

made ! mean axis1 ! pa?  ( ! inv>iable  
axis w be no long] a clos$ curve1 b an  
hyp]bola1 s t x w dep>t altoge!r f !    
neighb\rhood (  ! axle4 ,:5 ! top is 9  
?is condition x mu/ be spun g5tly1 =  x 
is v difficult to manage x :5 its motion
 gets m & m  ecc5tric4    ,:5 ! bob is  
screw$ /ill f>!r d[n1 ! axle becomes !  
axis  ( greate/ in]tia1 & xx#a ! lea/4  
,! major axis ( !  ellipse describ$ by !
 inv>iable axis w n[ be p]p5dicul> to   
xx#a1 & ! f>!r ! bob is screw$ d[n1 !   
ecc5tricity (  ! ellipse w dim9i%1 & !  
velocity ) :i* x is describ$  w         
increase4    ,i h n[ describ$ all !     
ph5om5a pres5t$ by a body revolv+       
freely on its c5tre ( gravity4 ,if we   
wi% to trace ! motion ( !  inv>iable    
axis by means ( ! col\r$ sectors1 we mu/
 make its  motion v sl[ comp>$ ) t ( !  
top4 ,x is necess>y1  !re=e1 to make !  
mom5ts ( in]tia ab\t ! pr9cipal axes v  
ne>ly equal1 & 9 ?is case a v small     
*ange 9 ! position (  any "p ( ! top w  
greatly d]ange ! position ( !  pr9cipal 

axis4 ,s t :5 ! top is well adju/$1 a   
s+le turn (  "o ( ! screws ( ! r+ is    
suffici5t to make ! axle no long]  a    
pr9cipal axis1 & to set ! true axis at a
 consid]able  incl9ation to ! axle ( !  
top4    ,all ! adju/m5ts mu/ !re=e be   
mo/ c>efully arrang$1 or we  may h !    
:ole app>atus d]ang$ by "s ecc5tricity (
 sp9n+4  ,! me?od ( mak+ ! pr9cipal axis
 co9cide ) ! axle mu/ be  /udi$ &       
practis$1 or ! fir/ attempt at sp9n+    
rapidly may end  9 ! de/ruction ( ! top1
 if n ! table on :i* x is spun4    ,on !
 ,e>?'s ,motion      ,we mu/ rememb] t  
^! motions ( a body ab\t its c5tre (    
gravity1 are n illu/rations ( ! !ory ( !
 precession ( !  ,equ9oxes4 ,precession 
c be illu/rat$ by ! app>atus1 b we mu/  
arrange x s t ! =ce ( gravity acts ! "p 
( ! attraction  ( ! sun & moon 9 produc+
 a =ce t5d+ to alt] ! axis (  rotation4 
,?is is easily done by br++ ! c5tre (   
gravity ( !  :ole a little bel[ ! po9t  
on :i* x sp9s4 ,! !ory ( su*  motions is
 f> m easily compreh5d$ ?an t :i* we h  

be5  inve/igat+4    ,b ! e>? is a body  
^: pr9cipal axes are unequal1 & f !     
ph5om5a ( precession we c det]m9e !     
ratio ( ! pol> &  equatorial axes ( !   
8c5tral ellipsoid20 & suppos+ ! e>? to  
h be5 set 9 motion ab\t any axis except 
! pr9cipal axis1 or to  h _h its orig9al
 axis di/urb$ 9 any way1 its subsequ5t  
motion  w\ld be t ( ! top :5 ! bob is a 
little bel[ ! critical  position4    ,! 
axis ( angul> mom5tum w\ld h an         
inv>iable position 9  space1 & w\ld     
travel ) respect to ! e>? r\nd ! axis ( 
figure ) a velocity display/yle         
omegafrac,c - ,a,a :]e  omega is !      
sid]eal angul> velocity ( ! e>?4 ,!     
app>5t  pole ( ! e>? w\ld travel 7)     
respect to ! e>?7 f we/ to  ea/ r\nd !  
true pole1 complet+ its circuit 9       
display/yle  frac,a,c - ,a sid]eal days1
 :i* appe>s to be ab\t #cbe4f sol>      
days4    ,! in/antane\s axis w\ld       
revolve ab\t ?is axis 9 space 9 ab\t  a 
day1 & w\ld always be 9 a plane ) ! true
 axis ( ! e>?  & ! axis ( angul>        

mom5tum4 ,! effect ( su* a motion on !  
app>5t position ( a /> w\ld be1 t its   
z5i? di/ance %\ld  be increas$ & dim9i%$
 dur+ a p]iod ( #cbe4f days4 ,?is       
alt]ation ( z5i? di/ance is ! same above
 & bel[ ! pole1 s  t ! pol> di/ance ( ! 
/> is unalt]$4 ,9 fact ! me?od (  f9d+ !
 pole ( ! heav5s by obs]vations ( />s1  
gives !  pole ( ! inv>iable axis1 :i* is
 alt]$ only by ext]nal  =ces1 su* z ^? (
 ! sun & moon4    ,!re is !re=e no *ange
 9 ! app>5t pol> di/ance ( />s  due to  
?is cause4 ,x is ! latitude :i* v>ies4  
,! magnitude (  ?is v>iation _c be      
det]m9$ by !ory4 ,! p]iodic "t ( !      
v>iation may be f\nd approximately f !  
kn[n dynamical  prop]ties ( ! e>?4 ,!   
epo* ( maximum latitude _c be f\nd      
except by obs]vation1 b x mu/ be lat] 9 
proportion to ! ea/  longitude ( !      
obs]vatory4    ,9 ord] to det]m9e !     
exi/5ce ( su* a v>iation ( latitude1 ,i 
h exam9$ ! obs]vations ( ,pol>is ) !    
,gre5wi* ,transit  ,circle 9 ! ye>s     
#ahea-b-c-d4 ,! obs]vations ( ! upp]    

transit  dur+ ea* mon? 7 collect$1 & !  
mean ( ea* mon? f\nd4 ,!  same 0 done = 
! l[] transits4 ,! diff]5ce ( z5i?      
di/ance  ( upp] & l[] transit is twice !
 pol> di/ance ( ,pol>is1 &  half ! sum  
gives ! co-latitude ( ,gre5wi*4    ,9   
?is way ,i f\nd ! app>5t co-latitude (  
,gre5wi* = ea*  mon? ( ! f\r ye>s       
specifi$4    ,!re appe>$ a v slight     
indication ( a maximum belong+ to !  set
 ( mon?s1      ,m>*1 #ea4 ,feb4 #eb4    
,dec4 #eb4 ,nov4 #ec4 ,sept4 #ed4       
,?is result1 h[ev]1 is to be reg>d$ z v 
d\btful1 z !re did  n appe> to be       
evid5ce = any v>iation exce$+ half a    
second (  space1 & m obs]vations w\ld be
 requir$ to e/abli% !  exi/5ce ( s small
 a v>iation at all4    ,i !re=e conclude
 t ! e>? has be5 = a long "t revolv+    
ab\t an axis v ne> to ! axis ( figure1  
if n co9cid+ )  x4 ,! cause ( ?is ne>   
co9cid5ce is ei!r ! orig9al s(tness  ( !
 e>?1 or ! pres5t fluidity ( its        
int]ior4 ,! axes ( !  e>? are s ne>ly   
equal1 t a consid]able elevation ( a    

tract (  c\ntry might produce a         
deviation ( ! pr9cipal axis )9 !  limits
 ( obs]vation1 & ! only cause :i* w\ld  
re/ore !  uni=m motion1 w\ld be ! action
 ( a fluid :i* w\ld gradually  dim9i% ! 
oscillations ( latitude4 ,! p]man5ce (  
latitude  ess5tially dep5ds on !        
inequality ( ! e>?'s axes1 = if !y  _h  
be5 all equal1 any alt]ation ( ! cru/ ( 
! e>? w\ld h  produc$ new pr9cipal axes1
 & ! axis ( rotation w\ld travel  ab\t  
^? axes1 alt]+ ! latitudes ( all places1
 & yet n 9  ! lea/ alt]+ ! position ( ! 
axis ( rotation among !  />s4    ,p]haps
 by a m ext5sive se>* & analysis ( !    
obs]vations (  diff]5t obs]vatories1 !  
nature ( ! p]iodic v>iation (  latitude1
 if x exi/1 may be det]m9$4 ,i am n aw>e
 ( any  calculations hav+ be5 made to   
prove its non-exi/5ce1 al?\gh1 on       
dynamical gr\nds1 we h e reason to look 
= "s v small  v>iation hav+ ! p]iodic "t
 ( #cbe4f days ne>ly1 a p]iod :i*  is   
cle>ly di/+ui%$ f any o!r a/ronomical   
cycle1 &  !re=e easily recognis$4       

,note3 ,dim5sions & ,weights ( ! p>ts ( 
! ,dynamical ,top4      ,"p ,weight lb4 
oz4 ,i4 ,body ( ! top-- ,mean diamet] ( 
r+1 #d  in*es4 ,section ( r+1 frac#ac   
in* squ>e4 ,! conical portion  rises f !
 upp] & inn] edge ( ! r+1 a height (    
#a;frac#ab in*es f ! base4 ,! :ole body 
( ! top  weighs #a #g ,ea* ( ! n9e      
adju/+ screws has its screw #a in*      
long1 & ! screw & head toge!r weigh #a  
ounce4 ,! :ole  weigh   #i ,,ii4 ,axle1 
c4-- ,l5g? ( axle #e in*es1 ( :i*       
frac#ab in* at ! bottom is occupi$ by ! 
/eel po9t1  #c;frac#ab in*es are brass )
 a good screw turn$ on x1 & !  rema9+   
in* is ( /eel1 ) a %>p po9t at ! top4 ,!
 :ole  weighs   #a;frac#ab ,! bob ,b has
 a diamet] ( #a4d in*es1  & a ?ickness (
 4#d4 ,x weighs   #b;frac#cd ,! nuts b  
& c1 = clamp+ ! bob & ! body ( ! top on 
! axle1 ea*  weigh frac#ab oz4   #a     
,weight ( :ole  top #b #e;frac#ad       
,! be/ arrangem5t1 = g5]al obs]vations1 
is to h ! disc (  c>d divid$ into f\r   
quadrants1 col\r$ ) v]milion1 *rome     

yell[1 em]ald gre51 & ultram>9e4 ,^! are
 bright col\rs1 &1  if ! v]milion is    
good1 !y comb9e into a grayi% t9t :5 !  
rotation is ab\t ! axle1 & bur/ into    
brilliant col\rs :5 !  axis is di/urb$4 
,x is useful to h "s conc5tric circles1 
drawn  ) ink1 ov] ! col\rs1 & ab\t #ab  
radii drawn 9 /rong p5cil  l9es4 ,x is  
easy to di/+ui% ! ink f ! p5cil l9es1 z 
!y  cross ! inv>iable axis1 by _! want (
 lu/re4 ,9 ?is way1 !  pa? ( ! inv>iable
 axis may be id5tifi$ ) great accuracy1 
&  comp>$ ) !ory4       99 #g? ,may     
#aheg4 ,! p>agraphs m>k$ ?us h be5      
rewritt5 s9ce !  pap] 0 read4