MR. S. S. HOUGH OH THE ROTATION OF AN ELASTIC SPHEROJD. 
339 
and therefore 
n — 
8-19 x 572 
522 
X 1 0 11 = 8-98 X 10 H . 
8 . Physical Characteristics of the Motion. 
The height of the waves at the free surface is given by the formula (29), viz. 
£ = — — e —rr (Oc,xz — d,yz). 
b a 1 + e e v 3 ' 
Comparing this with equation (18) we see that £ may be obtained from £ 0 by 
.66 
changing 0 V 0. z into 9 X - 0 2 -Thus, to our degree of approximation, the free 
surface will remain a spheroid of revolution of ellipticity e. The position of the axis 
6 
of this spheroid may be found by rotating the axis 0 z through small angles 9 1 --, 
6 + 6 
£ 
9o — , about Ox, 0 y respectively. 
Again, from (28) we see that with a high degree of approximation the displace¬ 
ments at the point x, y , 2 will be given by 
u — z 9. 2 , v — — z9 v iv = — x9. 2 -f- y0 1 .(32); 
in other words, the displacements due to elastic deformation will be negligible 
compared with the displacements due to the rotation of the body as a whole. We 
have seen, however, that the elastic distortions will produce an appreciable effect in 
displacing the axis of figure. 
The modified forms (32) indicate that 0 X , 0. 2 are the displacements of an axis 
sensibly fixed in the earth, which we may call the mean axis of figure, while the 
motion, at any instant, will consist very approximately of a rotation, as a rigid body, 
with angular velocity w about an axis) whose direction-cosines are 9fa, d 3 /w, 1. This 
axis we shall hereafter refer to as the instantaneous axis of rotation. It is, of course, 
only when we neglect the displacements due to elastic distortion that such an axis 
exists. 
We have found that when A = &» ( ^ 7 j, 0 1 = i0. 2 . Taking 
9 X = 
9 2 = - i<f>e iK{t ~C 
2x2 
where <£, r are real, we have 
