MR. S. S. HOUGH OH THE ROTATION OF AN ELASTIC SPHEROID. 
341 
the axis of rotation remains sensibly fixed in direction while the mean axis of figure 
and the axis of the distorted figure describe cones of revolution about it in a period 
approximately equal to the period of rotation, these three axes always remaining in 
a plane. 
Let us next consider the motion of the instantaneous axis of rotation relatively to 
the Earth. Take a new set of axes Ox lt O y x , O 24 found by rotating the old set 
through small angles 0 V 0. 2 about Ox, O y ; these new axes will be sensibly fixed in 
the Earth. The motion at any instant consists of a rotation whose components a,bout 
the old axes are 9 V 0 2 , co ; resolving these rotations about the new axes, we find for 
the components 
~~ ojO.-), e 0 -J- (o0 j, <o. 
Hence the direction-cosines of the axis of rotation relatively to the axes Oaq, Oy l} O z 1 
are 
(6y (O0. 2 )/(O , ( 0 -f- (oO-^jdJ, 1, 
or 
“ + ;r^ 7 ,)sin\(« ~ r), + 2 0^1 + - r), 1. 
Thus, relatively to the Earth, the axis of rotation will describe a cone of revolution 
about the mean axis of figure with angular velocity X, the direction of motion in 
this cone being the same as the direction of the Earth's rotation. This motion would 
manifest itself by a periodic change in the latitude of places on the Earth’s surface, 
as found by astronomical observations. 
The circumstances we have here described are the well-known characteristics of the 
Eulerian nutation,* with the additional feature that the axis of figure is displaced by 
centrifugal force towards the axis of rotation. This latter fact has been assumed by 
Newcomb! and other writers^ as the basis of their work. The law of displacement 
of the pole of figure assumed by Newcomb is, however, not verified. We find 
PR : PP'R = e + e' : e, 
or 
PP':P'R = e :e . (a), 
whereas Newcomb has taken 
PP' : P'R = E' : e.( 6 ), 
where E' denotes the ellipticity which would be induced in a sphere by centrifugal 
* Tisserand, ‘ Mecanique Celeste,’ vol. 2, p. 494. 
t ‘ Monthly Notices,’ March, 1892. 
| For an account of previous work, vide Tisserand, ‘ Mecanique Celeste,’ vol. 2, chap. xxx. 
