,on a ,dynamical ,top1      for         
exhibiting the phenomena of the motion  
of a system of invariable  form about a 
fixed point1 with some suggestions as to
 the ,earth's  motion  ,james ,clerk    
,maxwell    ,from the ,transactions of  
the ,royal ,society of ,edinburgh1 ,vol4
 ,,xxi4  ,part ,,iv4  7,read #bjth      
,april1 #aheg47      ,to those who study
 the progress of exact science1 the     
common  spinning-top is a symbol of the 
labours and the perplexities of men who 
had successfully threaded the mazes of  
the planetary motions4 ,the             
mathematicians of the last age1         
searching through nature for problems   
worthy of their analysis1 found in this 
toy of their youth1 ample  occupation   
for their highest mathematical powers4  
,no illustration of astronomical        
precession can be devised more perfect  
than that presented by a properly       
balanced top1 but yet the motion of     
rotation has intricacies far exceeding  
those of the theory of  precession4     

,accordingly1 we find ,euler and        
,d',alembert devoting their talent and  
their patience to the establishment of  
the laws of the rotation of  solid      
bodies4 ,lagrange has incorporated his  
own analysis of the problem  with his   
general treatment of mechanics1 and     
since his time ,m4 ,poinst  has brought 
the subject under the power of a more   
searching analysis  than that of the    
calculus1 in which ideas take the place 
of symbols1  and intelligible           
propositions supersede equations4    ,in
 the practical department of the        
subject1 we must notice the rotatory    
machine of ,bohnenberger1 and the       
nautical top of ,troughton4 ,in the     
first of these instruments we have the  
model of the ,gyroscope1 by which       
,foucault has been able to render       
visible the effects of the earth's      
rotation4 ,the beautiful experiments by 
which ,mr ,j4 ,elliot has made the      
ideas of precession so familiar to us   
are performed with a top1 similar  in   

some respects to ,troughton's1 though   
not borrowed from his4    ,the top which
 ,i have the honour to spin before the  
,society1 differs  from that of ,mr     
,elliot in having more adjustments1 and 
in being  designed to exhibit far more  
complicated phenomena4    ,the          
arrangement of these adjustments1 so as 
to produce the desired  effects1 depends
 on the mathematical theory of rotation4
 ,the method of  exhibiting the motion  
of the axis of rotation1 by means of a  
coloured  disc1 is essential to the     
success of these adjustments4 ,this     
optical  contrivance for rendering      
visible the nature of the rapid motion  
of the  top1 and the practical methods  
of applying the theory of rotation to   
such an instrument as the one before us1
 are the grounds on which ,i  bring my  
instrument and experiments before the   
,society as my own4    ,i propose1      
therefore1 in the first place1 to give a
 brief outline of  such parts of the    
theory of rotation as are necessary for 

the  explanation of the phenomena of the
 top4    ,i shall then describe the     
instrument with its adjustments1 and the
  effect of each1 the mode of observing 
of the coloured disc when the top  is in
 motion1 and the use of the top in      
illustrating the mathematical  theory1  
with the method of making the different 
experiments4    ,lastly1 ,i shall       
attempt to explain the nature of a      
possible variation  in the earth's axis 
due to its figure4 ,this variation1 if  
it exists1  must cause a periodic       
inequality in the latitude of every     
place on the  earth's surface1 going    
through its period in about eleven      
months4 ,the  amount of variation must  
be very small1 but its character gives  
it  importance1 and the necessary       
observations are already made1 and only 
require reduction4    ,on the ,theory of
 ,rotation4      ,the theory of the     
rotation of a rigid system is strictly  
deduced from  the elementary laws of    
motion1 but the complexity of the motion

 of the  particles of a body freely     
rotating renders the subject so         
intricate1  that it has never been      
thoroughly understood by any but the    
most expert  mathematicians4 ,many who  
have mastered the lunar theory have come
 to  erroneous conclusions on this      
subject2 and even ,newton has chosen to 
deduce the disturbance of the earth's   
axis from his theory of the  motion of  
the nodes of a free orbit1 rather than  
attack the problem of  the rotation of a
 solid body4    ,the method by which ,m4
 ,poinst has rendered the theory more   
manageable1  is by the liberal          
introduction of 8appropriate ideas10    
chiefly of a  geometrical character1    
most of which had been rendered familiar
 to  mathematicians by the writings of  
,monge1 but which then first became     
illustrations of this branch of         
dynamics4 ,if any further progress is to
  be made in simplifying and arranging  
the theory1 it must be by the  method   
which ,poinst has repeatedly pointed out

 as the only one which  can lead to a   
true knowledge of the subject1--that of 
proceeding from  one distinct idea to   
another instead of trusting to symbols  
and  equations4    ,an important        
contribution to our stock of appropriate
 ideas and methods  has lately been made
 by ,mr ,r4 ,b4 ,hayward1 in a paper1   
8,on a ,direct  ,method of estimating   
,velocities1 ,accelerations1 and all    
similar  quantities1 with respect to    
axes1 moveable in any manner in ,space40
  7,trans4 ,cambridge ,phil4 ,soc ,vol4 
x4 ,part ,i47    99 ,in this            
communication ,i intend to confine      
myself to that part of the  subject     
which the top is intended io illustrate1
 namely1 the alteration  of the position
 of the axis in a body rotating freely  
about its centre  of gravity4 ,i shall1 
therefore1 deduce the theory as briefly 
as  possible1 from two considerations   
only1--the permanence of the original   
angular momentum in direction and       
magnitude1 and the permanence of  the   

original vis viva4    99 ,the           
mathematical difficulties of the theory 
of rotation arise chiefly  from the want
 of geometrical illustrations and       
sensible images1 by  which we might fix 
the results of analysis in our minds4   
,it is easy to understand the motion of 
a body revolving about a fixed  axle4   
,every point in the body describes a    
circle about the axis1 and  returns to  
its original position after each        
complete revolution4 ,but if  the axle  
itself be in motion1 the paths of the   
different points of the  body will no   
longer be circular or re-entrant4 ,even 
the velocity of  rotation about the axis
 requires a careful definition1 and the 
proposition that1 in all motion about a 
fixed point1 there is always  one line  
of particles forming an instantaneous   
axis1 is usually given  in the form of a
 very repulsive mass of calculation4    
,most of these  difficulties may be got 
rid of by devoting a little attention to
 the  mechanics and geometry of the     

problem before entering on the          
discussion  of the equations4    ,mr    
,hayward1 in his paper already referred 
to1 has made great use of the           
mechanical conception of ,angular       
,momentum4      ,definition #a ,the     
,angular ,momentum of a particle about  
an axis is  measured by the product of  
the mass of the particle1 its velocity  
resolved in the normal plane1 and the   
perpendicular from the axis on  the     
direction of motion4    99 ,the angular 
momentum of any system about an axis is 
the algebraical  sum of the angular     
momenta of its parts4    ,as the rate of
 change of the linear momentum of a     
particle measures  the moving force     
which acts on it1 so the rate of change 
of  angular momentum measures the moment
 of that force about an axis4    ,all   
actions between the parts of a system1  
being pairs of equal and  opposite      
forces1 produce equal and opposite      
changes in the angular  momentum of     
those parts4 ,hence the whole angular   

momentum of the system  is not affected 
by these actions and re-actions4    99  
,when a system of invariable form       
revolves about an axis1 the angular     
velocity of every part is the same1 and 
the angular momentum about the  axis is 
the product of the angular velocity and 
the moment of  inertia about that axis4 
99 ,it is only in particular cases1     
however1 that the whole angular         
momentum can be estimated in this way4  
,in general1 the axis of angular        
momentum differs from the axis of       
rotation1 so that there will be a       
residual angular momentum about an axis 
perpendicular to that of  rotation1     
unless that axis has one of three       
positions1 called the  principal axes of
 the body4    ,by referring everything  
to these three axes1 the theory is      
greatly  simplified4 ,the moment of     
inertia about one of these axes is      
greater  than that about any other axis 
through the same point1 and that about  
one of the others is a minimum4 ,these  

two are at right angles1 and the  third 
axis is perpendicular to their plane1   
and is called the mean  axis4    99 ,let
 ,a1 ,b1 ,c be the moments of inertia   
about the principal axes  through the   
centre of gravity1 taken in order of    
magnitude1 and let  omega#a omega#b     
omega#c be the angular velocities about 
them1  then the angular momenta will be 
,aomega#a1 ,bomega#b1 and  ,comega#c4   
,angular momenta may be compounded like 
forces or velocities1 by the law  of the
 8parallelogram10 and since these three 
are at right angles to  each other1     
their resultant is      begindisplaymath
 sqrt,a#bomega#ab  ,b#bomega#bb         
,c#bomega#cb  ,h enddisplaymath 7#a7    
and this must be constant1 both in      
magnitude and direction in space1  since
 no external forces act on the body4    
,we shall call this axis of angular     
momentum the invariable axis4 ,it  is   
perpendicular to what has been called   
the invariable plane4 ,poinst  calls it 
the axis of the couple of impulsion4    

,the direction-cosines  of this axis in 
the body are1      begindisplaymath     
beginarrayc c c displaystyle l          
frac,aomega#a;,h1 444 444ga#b;,h1       
displaystyle n   frac,comega#c;,h4      
endarrayenddisplaymath      ,since ,i1 m
 and n vary during the motion1 we need  
some additional  condition to determine 
the relation between them4 ,we find this
 in the  property of the vis viva of a  
system of invariable form in which      
there is no friction4 ,the vis viva of  
such a system must be constant4  ,we    
express this in the equation            
begindisplaymath ,aomega#ab  ,bomega#bb 
,comega#cb  ,v  enddisplaymath 7#b7     
,substituting the values of omega#a1    
omega#b1 omega#c in terms  of l1 m1 n1  
begindisplaymath fracl#b;,a  fracm#b;,b 
fracn#b;,c   frac,v,h#b4 enddisplaymath 
,let #a_/;,a  a#b1 #a_/;,b  b#b1 #a_/;c 
c#b1 ,v_/,h#b  e#b1 and this  equation  
becomes      begindisplaymath a#bl#b    
b#bm#b  c#bn#b  e#b  enddisplaymath 7#c7
    and the equation to the cone1       

described by the invariable axis within 
the body1 is      begindisplaymath 7a#b 
- e#b7 x#b  7b#b - e#b7 y#b  7c#b - e#b7
 z#b   #j enddisplaymath 7#d7      ,the 
intersections of this cone with planes  
perpendicular to the  principal axes are
 found by putting x1 y1 or z1 constant  
in this  equation4 ,by giving e various 
values1 all the different paths of the  
pole of the invariable axis1            
corresponding to different initial      
circumstances1 may be traced4           
,figure3 ,figure #a      99 ,in the     
figures1 ,i have supposed a#b  #ajj1 b#b
 #ajg1 and c#b  #aaj4 ,the first figure 
represents a section of the various     
cones by a  plane perpendicular to the  
axis of x1 which is that of greatest    
moment of inertia4 ,these sections are  
ellipses having their major axis        
parallel to the axis of b4 ,the value of
 e#b corresponding to each  of these    
curves is indicated by figures beside   
the curve4 ,the  ellipticity increases  
with the size of the ellipse1 so that   

the section  corresponding to e#b  #ajg 
would be two parallel straight lines    
7beyond the bounds of the figure71 after
 which the sections would be            
hyperbolas4      ,figure3 ,figure #b    
99 ,the second figure represents the    
sections made by a plane1  perpendicular
 to the mean axis4 ,they are all        
hyperbolas1 except when  e#b  #ajg1 when
 the section is two intersecting        
straight lines4      ,figure3 ,figure #c
      ,the third figure shows the       
sections perpendicular to the axis of   
least  moment of inertia4 ,from e#b     
#aaj to e#b  #ajg the sections are      
ellipses1 e#b  #ajg gives two parallel  
straight lines1 and beyond  these the   
curves are hyperbolas4      ,figure3    
,figure #d      99 ,the fourth and fifth
 figures show the sections of the series
 of cones  made by a cube and a sphere  
respectively4 ,the use of these figures 
is  to exhibit the connexion between the
 different curves described about  the  
three principal axes by the invariable  

axis during the motion of  the body4    
,figure3 ,figure #e      99 ,we have    
next to compare the velocity of the     
invariable axis with  respect to the    
body1 with that of the body itself round
 one of the  principal axes4 ,since the 
invariable axis is fixed in space1 its  
motion  relative to the body must be    
equal and opposite to that of the       
portion  of the body through which it   
passes4 ,now the angular velocity of a  
portion of the body whose               
direction-cosines are l1 m1 n1 about    
the axis of x is      begindisplaymath  
fracomega#aa - l#b - fracl#a -          
l#b7lomega#a  momega#b  nomega-#c74     
enddisplaymath      ,substituting the   
values of omega#a1 omega#b1 omega#c1 in 
terms  of l1 m1 n1 and taking account of
 equation 7#c71 this expression  becomes
      begindisplaymath ,hfrac7a#b -     
e#b7a - l#bl4 enddisplaymath            
,changing the sign and putting          
displaystyle l  fracomega#a;a#b;,h  we  
have the angular velocity of the        

invariable axis about that of x         
begindisplaymath  fracomega#aa - l#b    
frace#b - a#b;a#b1  enddisplaymath      
always positive about the axis of       
greatest moment1 negative about that  of
 least moment1 and positive or negative 
about the mean axis according  to the   
value of e#b4 ,the direction of the     
motion in every case is  represented by 
the arrows in the figures4 ,the arrows  
on the outside of  each figure indicate 
the direction of rotation of the body4  
99 ,if we attend to the curve described 
by the pole of the invariable  axis on  
the sphere in fig4 #e1 we shall see that
 the areas described by  that point1 if 
projected on the plane of yz1 are swept 
out at the  rate      begindisplaymath  
omega#a frace#b - a#b;a#b4              
enddisplaymath      ,now the semi-axes  
of the projection of the spherical      
ellipse described  by the pole are      
begindisplaymath sqrtfrace#b - a#b;b#b -
 a#b  hspace#a;cmtextrmandhspace#a;cm   
sqrtfrace#b - a#b;c#b -  a#b4           

enddisplaymath      ,dividing the area  
of this ellipse by the area described   
during one  revolution of the body1 we  
find the number of revolutions of the   
body  during the description of the     
ellipse--      begindisplaymath         
fraca#bsqrtb#b - a#bsqrtc#b - a#b4      
enddisplaymath      ,the projections of 
the spherical ellipses upon the plane of
 yz are  all similar ellipses1 and      
described in the same number of         
revolutions2  and in each ellipse so    
projected1 the area described in any    
time is  proportional to the number of  
revolutions of the body about the axis  
of  x1 so that if we measure time by    
revolutions of the body1 the motion  of 
the projection of the pole of the       
invariable axis is identical with  that 
of a body acted on by an attractive     
central force varying directly  as the  
distance4 ,in the case of the hyperbolas
 in the plane of the  greatest and least
 axis1 this force must be supposed      
repulsive4 ,the  dots in the figures #a1

 #b1 #c1 are intended to indicate       
roughly the  progress made by the       
invariable axis during each revolution  
of the body  about the axis of x1 y and 
z respectively4 ,it must be remembered  
that the rotation about these axes      
varies with their inclination to the    
invariable axis1 so that the angular    
velocity diminishes as the  inclination 
increases1 and therefore the areas in   
the ellipses above  mentioned are not   
described with uniform velocity in      
absolute time1 but  are less rapidly    
swept out at the extremities of the     
major axis than at  those of the minor4 
99 ,when two of the axes have equal     
moments of inertia1 or b  c1 then  the  
angular velocity omega#a is constant1   
and the path of the  invariable axis is 
circular1 the number of revolutions of  
the body  during one circuit of the     
invariable axis1 being                  
begindisplaymath fraca#b;b#b - a#b      
enddisplaymath      ,the motion is in   
the same direction as that of the       

rotation1 or in the  opposite direction1
 according as the axis of x is that of  
greatest or  of least moment of inertia4
    99 ,both in this case1 and in that  
in which the three axes are unequal1    
the motion of the invariable axis in the
 body may be rendered very slow  by     
diminishing the difference of the       
moments of inertia4 ,the angular        
velocity of the axis of x about the     
invariable axis in space is             
begindisplaymath omega#a;frace#b -      
a#bl#b;a#b7a - l#b71  enddisplaymath    
which is greater or less than omega#a1  
as e#b is greater or less  than a#b1    
and1 when these quantities are nearly   
equal1 is very nearly  the same as      
omega#a itself4 ,this quantity indicates
 the rate of  revolution of the axle of 
the top about its mean position1 and is 
very  easily observed4    99 ,the       
instantaneous axis is not so easily     
observed4 ,it revolves round  the       
invariable axis in the same time with   
the axis of x1 at a  distance which is  

very small in the case when a1 b1 c1 are
 nearly  equal4 ,from its rapid angular 
motion in space1 and its near           
coincidence  with the invariable axis1  
there is no advantage in studying its   
motion  in the top4    99 ,by making the
 moments of inertia very unequal1 and in
 definite  proportion to each other1 and
 by drawing a few strong lines as       
diameters of the disc1 the combination  
of motions will produce an  appearance  
of epicycloids1 which are the result of 
the continued  intersection of the      
successive positions of these lines1 and
 the cusps  of the epicycloids lie in   
the curve in which the instantaneous    
axis  travels4 ,some of the figures     
produced in this way are very pleasing4 
,in order to illustrate the theory of   
rotation experimentally1 we must  have a
 body balanced on its centre of gravity1
 and capable of having  its principal   
axes and moments of inertia altered in  
form and position  within certain       
limits4 ,we must be able to make the    

axle of the  instrument the greatest1   
least1 or mean principal axis1 or to    
make it  not a principal axis at all1   
and we must be able to see the position 
of the invariable axis of rotation at   
any time4 ,there must be three          
adjustments to regulate the position of 
the centre of gravity1 three  for the   
magnitudes of the moments of inertia1   
and three for the  directions of the    
principal axes1 nine independent        
adjustments1 which  may be distributed  
as we please among the screws of the    
instrument4      ,figure3 ,figure #f    
,the form of the body of the instrument 
which ,i have found most suitable  is   
that of a bell 7fig4 #f74 ,c is a hollow
 cone of brass1 ,r is a  heavy ring cast
 in the same piece4 ,six screws1 with   
heavy heads1 x1  y1 z1 x'1 y'1 z'1 work 
horizontally in the ring1 and three     
similar screws1 l1 m1 n1 work vertically
 through the ring at  equal intervals4  
,,as is the axle of the instrument1 ,,ss
 is a brass  screw working in the upper 

part of the cone ,c1 and capable of     
being  firmly clamped by means of the   
nut c4 ,b is a cylindrical brass bob1   
which may be screwed up or down the     
axis1 and fixed in its place by the  nut
 b4    ,the lower extremity of the axle 
is a fine steel point1 finished without 
emery1 and afterwards hardened4 ,it runs
 in a little agate cup set in  the top  
of the pillar ,p4 ,if any emery had been
 embedded in the steel1  the cup would  
soon be worn out4 ,the upper end of the 
axle has also a  steel point by which it
 may be kept steady while spinning4     
,when the instrument is in use1 a       
coloured disc is attached to the upper  
end of the axle4    ,it will be seen    
that there are eleven adjustments1 nine 
screws in the  brass ring1 the axle     
screwing in the cone1 and the bob       
screwing on the  axle4 ,the advantage of
 the last two adjustments is1 that by   
them large  alterations can be made1    
which are not possible by means of the  
small  screws4    ,the first thing to be

 done with the instrument is1 to make   
the steel  point at the end of the axle 
coincide with the centre of gravity of  
the  whole4 ,this is done roughly by    
screwing the axle to the right place    
nearly1 and then balancing the          
instrument on its point1 and screwing   
the bob and the horizontal screws till  
the instrument will remain  balanced in 
any position in which it is placed4     
,when this adjustment is carefully made1
 the rotation of the top has no         
tendency to shake the steel point in the
 agate cup1 however irregular  the      
motion may appear to be4    ,the next   
thing to be done1 is to make one of the 
principal axes of the  central ellipsoid
 coincide with the axle of the top4     
,to effect this1 we must begin by       
spinning the top gently about its  axle1
 steadying the upper part with the      
finger at first4 ,if the axle is        
already a principal axis the top will   
continue to revolve about its  axle when
 the finger is removed4 ,if it is not1  

we observe that the top  begins to spin 
about some other axis1 and the axle     
moves away from the  centre of motion   
and then back to it again1 and so on1   
alternately  widening its circles and   
contracting them4    ,it is impossible  
to observe this motion successfully1    
without the aid  of the coloured disc   
placed near the upper end of the axis4  
,this disc  is divided into sectors1 and
 strongly coloured1 so that each sector 
may  be recognised by its colour when in
 rapid motion4 ,if the axis about  which
 the top is really revolving1 falls     
within this disc1 its position  may be  
ascertained by the colour of the spot at
 the centre of motion4  ,if the central 
spot appears red1 we know that the      
invariable axis at  that instant passes 
through the red part of the disc4    ,in
 this way we can trace the motion of the
 invariable axis in the  revolving body1
 and we find that the path which it     
describes upon the  disc may be a       
circle1 an ellipse1 an hyperbola1 or a  

straight line1  according to the        
arrangement of the instrument4    ,in   
the case in which the invariable axis   
coincides at first with the  axle of the
 top1 and returns to it after separating
 from it for a time1  its true path is a
 circle or an ellipse having the axle in
 its  circumference4 ,the true principal
 axis is at the centre of the closed    
curve4 ,it must be made to coincide with
 the axle by adjusting the  vertical    
screws l1 m1 n4    ,suppose that the    
colour of the centre of motion1 when    
farthest from the  axle1 indicated that 
the axis of rotation passed through the 
sector  ,l1 then the principal axis must
 also lie in that sector at half the    
distance from the axle4    ,if this     
principal axis be that of greatest      
moment of inertia1 we must  raise the   
screw l in order to bring it nearer the 
axle ,a4 ,if it  be the axis of least   
moment we must lower the screw l4 ,in   
this way  we may make the principal axis
 coincide with the axle4 ,let us suppose

  that the principal axis is that of    
greatest moment of inertia1 and that  we
 have made it coincide with the axle of 
the instrument4 ,let us also  suppose   
that the moments of inertia about the   
other axes are equal1 and  very little  
less than that about the axle4 ,let the 
top be spun about  the axle and then    
receive a disturbance which causes it to
 spin about  some other axis4 ,the      
instantaneous axis will not remain at   
rest either  in space or in the body4   
,in space it will describe a right cone1
  completing a revolution in somewhat   
less than the time of revolution of  the
 top4 ,in the body it will describe     
another cone of larger angle in a       
period which is longer as the difference
 of axes of the body is  smaller4 ,the  
invariable axis will be fixed in space1 
and describe a  cone in the body4       
,the relation of the different motions  
may be understood from the  following   
illustration4 ,take a hoop and make it  
revolve about a stick  which remains at 

rest and touches the inside of the hoop4
 ,the section  of the stick represents  
the path of the instantaneous axis in   
space1  the hoop that of the same axis  
in the body1 and the axis of the stick  
the invariable axis4 ,the point of      
contact represents the pole of the      
instantaneous axis itself1 travelling   
many times round the stick before  it   
gets once round the hoop4 ,it is easy to
 see that the direction in  which the   
hoop moves round the stick1 so that if  
the top be spinning in  the direction   
,l1 ,m1 ,n1 the colours will appear in  
the same order4    ,by screwing the bob 
,b up the axle1 the difference of the   
axes of  inertia may be diminished1 and 
the time of a complete revolution of the
  invariable axis in the body increased4
 ,by observing the number of            
revolutions of the top in a complete    
cycle of colours of the invariable      
axis1 we may determine the ratio of the 
moments of inertia4    ,by screwing the 
bob up farther1 we may make the axle the

 principal axis  of least moment of     
inertia4    ,the motion of the          
instantaneous axis will then be that of 
the point of  contact of the stick with 
the outside of the hoop rolling on it4  
,the  order of colours will be ,n1 ,m1  
,l1 if the top be spinning in the       
direction ,l1 ,m1 ,n1 and the more the  
bob is screwed up1 the more  rapidly    
will the colours change1 till it ceases 
to be possible to make  the observations
 correctly4    ,in calculating the      
dimensions of the parts of the          
instrument1 it is  necessary to provide 
for the exhibition of the instrument    
with its axle  either the greatest or   
the least axis of inertia4 ,the         
dimensions and  weights of the parts of 
the top which ,i have found most        
suitable1 are  given in a note at the   
end of this paper4    ,now let us make  
the axes of inertia in the plane of the 
ring unequal4  ,we may do this by       
screwing the balance screws x and x#a   
farther  from the axle without altering 

the centre of gravity4    ,let us       
suppose the bob ,b screwed up so as to  
make the axle the axis  of least        
inertia4 ,then the mean axis is parallel
 to xx#a1 and the  greatest is at right 
angles to xx#a in the horizontal plane4 
,the path  of the invariable axis on the
 disc is no longer a circle but an      
ellipse1 concentric with the disc1 and  
having its major axis parallel  to the  
mean axis xx#a4    ,the smaller the     
difference between the moment of inertia
 about the axle  and about the mean     
axis1 the more eccentric the ellipse    
will be2 and  if1 by screwing the bob   
down1 the axle be made the mean axis1   
the path  of the invariable axis will be
 no longer a closed curve1 but an       
hyperbola1 so that it will depart       
altogether from the neighbourhood of    
the axle4 ,when the top is in this      
condition it must be spun gently1 for   
it is very difficult to manage it when  
its motion gets more and more           
eccentric4    ,when the bob is screwed  

still farther down1 the axle becomes the
 axis  of greatest inertia1 and xx#a the
 least4 ,the major axis of the  ellipse 
described by the invariable axis will   
now be perpendicular to  xx#a1 and the  
farther the bob is screwed down1 the    
eccentricity of  the ellipse will       
diminish1 and the velocity with which it
 is described  will increase4    ,i have
 now described all the phenomena        
presented by a body revolving  freely on
 its centre of gravity4 ,if we wish to  
trace the motion of the  invariable axis
 by means of the coloured sectors1 we   
must make its  motion very slow compared
 with that of the top4 ,it is necessary1
  therefore1 to make the moments of     
inertia about the principal axes very   
nearly equal1 and in this case a very   
small change in the position of  any    
part of the top will greatly derange the
 position of the  principal axis4 ,so   
that when the top is well adjusted1 a   
single turn of  one of the screws of the
 ring is sufficient to make the axle no 

longer  a principal axis1 and to set the
 true axis at a considerable            
inclination to the axle of the top4     
,all the adjustments must therefore be  
most carefully arranged1 or we  may have
 the whole apparatus deranged by some   
eccentricity of spinning4  ,the method  
of making the principal axis coincide   
with the axle must be  studied and      
practised1 or the first attempt at      
spinning rapidly may end  in the        
destruction of the top1 if not the table
 on which it is spun4    ,on the        
,earth's ,motion      ,we must remember 
that these motions of a body about its  
centre of  gravity1 are not             
illustrations of the theory of the      
precession of the  ,equinoxes4          
,precession can be illustrated by the   
apparatus1 but we must  arrange it so   
that the force of gravity acts the part 
of the attraction  of the sun and moon  
in producing a force tending to alter   
the axis of  rotation4 ,this is easily  
done by bringing the centre of gravity  

of the  whole a little below the point  
on which it spins4 ,the theory of such  
motions is far more easily comprehended 
than that which we have been            
investigating4    ,but the earth is a   
body whose principal axes are unequal1  
and from the  phenomena of precession we
 can determine the ratio of the polar   
and  equatorial axes of the 8central    
ellipsoid20 and supposing the earth to  
have been set in motion about any axis  
except the principal axis1 or to  have  
had its original axis disturbed in any  
way1 its subsequent motion  would be    
that of the top when the bob is a little
 below the critical  position4    ,the  
axis of angular momentum would have an  
invariable position in  space1 and would
 travel with respect to the earth round 
the axis of  figure with a velocity     
displaystyle  omegafrac,c - ,a,a where  
omega is the sidereal angular velocity  
of the earth4 ,the apparent  pole of the
 earth would travel 7with respect to the
 earth7 from west to  east round the    

true pole1 completing its circuit in    
displaystyle  frac,a,c - ,a sidereal    
days1 which appears to be about #cbe4f  
solar  days4    ,the instantaneous axis 
would revolve about this axis in space  
in about  a day1 and would always be in 
a plane with the true axis of the earth 
and the axis of angular momentum4 ,the  
effect of such a motion on the  apparent
 position of a star would be1 that its  
zenith distance should  be increased and
 diminished during a period of #cbe4f   
days4 ,this  alteration of zenith       
distance is the same above and below the
 pole1 so  that the polar distance of   
the star is unaltered4 ,in fact the     
method of  finding the pole of the      
heavens by observations of stars1 gives 
the  pole of the invariable axis1 which 
is altered only by external  forces1    
such as those of the sun and moon4      
,there is therefore no change in the    
apparent polar distance of stars  due to
 this cause4 ,it is the latitude which  
varies4 ,the magnitude of  this         

variation cannot be determined by       
theory4 ,the periodic time of the       
variation may be found approximately    
from the known dynamical  properties of 
the earth4 ,the epoch of maximum        
latitude cannot be found  except by     
observation1 but it must be later in    
proportion to the east  longitude of the
 observatory4    ,in order to determine 
the existence of such a variation of    
latitude1 ,i  have examined the         
observations of ,polaris with the       
,greenwich ,transit  ,circle in the     
years #ahea-b-c-d4 ,the observations of 
the upper transit  during each month    
were collected1 and the mean of each    
month found4 ,the  same was done for the
 lower transits4 ,the difference of     
zenith distance  of upper and lower     
transit is twice the polar distance of  
,polaris1 and  half the sum gives the   
co-latitude of ,greenwich4    ,in this  
way ,i found the apparent co-latitude of
 ,greenwich for each  month of the four 
years specified4    ,there appeared a   

very slight indication of a maximum     
belonging to the  set of months1        
,march1 #ea4 ,feb4 #eb4 ,dec4 #eb4 ,nov4
 #ec4 ,sept4 #ed4      ,this result1    
however1 is to be regarded as very      
doubtful1 as there did  not appear to be
 evidence for any variation exceeding   
half a second of  space1 and more       
observations would be required to       
establish the  existence of so small a  
variation at all4    ,i therefore       
conclude that the earth has been for a  
long time revolving  about an axis very 
near to the axis of figure1 if not      
coinciding with  it4 ,the cause of this 
near coincidence is either the original 
softness  of the earth1 or the present  
fluidity of its interior4 ,the axes of  
the  earth are so nearly equal1 that a  
considerable elevation of a tract of    
country might produce a deviation of the
 principal axis within the  limits of   
observation1 and the only cause which   
would restore the  uniform motion1 would
 be the action of a fluid which would   

gradually  diminish the oscillations of 
latitude4 ,the permanence of latitude   
essentially depends on the inequality of
 the earth's axes1 for if they  had been
 all equal1 any alteration of the crust 
of the earth would have  produced new   
principal axes1 and the axis of rotation
 would travel  about those axes1        
altering the latitudes of all places1   
and yet not in  the least altering the  
position of the axis of rotation among  
the  stars4    ,perhaps by a more       
extensive search and analysis of the    
observations of  different              
observatories1 the nature of the        
periodic variation of  latitude1 if it  
exist1 may be determined4 ,i am not     
aware of any  calculations having been  
made to prove its non-existence1        
although1 on  dynamical grounds1 we have
 every reason to look for some very     
small  variation having the periodic    
time of #cbe4f days nearly1 a period    
which  is clearly distinguished from any
 other astronomical cycle1 and          

therefore easily recognised4    ,note3  
,dimensions and ,weights of the parts of
 the ,dynamical ,top4      ,part ,weight
 lb4 oz4 ,i4 ,body of the top-- ,mean   
diameter of ring1 #d  inches4 ,section  
of ring1 frac#ac inch square4 ,the      
conical portion  rises from the upper   
and inner edge of the ring1 a height of 
#a;frac#ab inches from the base4 ,the   
whole body of the top  weighs #a #g     
,each of the nine adjusting screws has  
its screw #a inch  long1 and the screw  
and head together weigh #a ounce4 ,the  
whole  weigh   #i ,,ii4 ,axle1 c4--     
,length of axle #e inches1 of which     
frac#ab inch at the bottom is occupied  
by the steel point1  #c;frac#ab inches  
are brass with a good screw turned on   
it1 and the  remaining inch is of steel1
 with a sharp point at the top4 ,the    
whole  weighs   #a;frac#ab ,the bob ,b  
has a diameter of #a4d inches1  and a   
thickness of 4#d4 ,it weighs            
#b;frac#cd ,the nuts b  and c1 for      
clamping the bob and the body of the top

 on the axle1 each  weigh frac#ab oz4   
#a ,weight of whole  top #b #e;frac#ad  
,the best arrangement1 for general      
observations1 is to have the disc of    
card divided into four quadrants1       
coloured with vermilion1 chrome  yellow1
 emerald green1 and ultramarine4 ,these 
are bright colours1 and1  if the        
vermilion is good1 they combine into a  
grayish tint when the  rotation is about
 the axle1 and burst into brilliant     
colours when the  axis is disturbed4 ,it
 is useful to have some concentric      
circles1 drawn  with ink1 over the      
colours1 and about #ab radii drawn in   
strong pencil  lines4 ,it is easy to    
distinguish the ink from the pencil     
lines1 as they  cross the invariable    
axis1 by their want of lustre4 ,in this 
way1 the  path of the invariable axis   
may be identified with great accuracy1  
and  compared with theory4       99 #gth
 ,may #aheg4 ,the paragraphs marked thus
 have been rewritten since the  paper   
was read4