340 MR. S. S. HOUGH OH THE ROTATION OF AN ELASTIC SPHEROID. 
Adding to these the solutions obtained by changing the sign of i, wherever it 
occurs, we find as the real solution corresponding to the frequencies fir -, 
6 X = 2(f) cos X (t — r), 
2(f) 
-— sin X (t — t), 
CO 
0 2 = 2(f) sin A .(t — t), 
+ 2(f) —- cos \(t — t). 
y 
Let C be the point where the axis Oz cuts the surface of the spheroid, and let us 
take a pair of rectangular axes Cx, C y, having C as origin and parallel to the original 
axes Ox, 0 y. Let P, P 7 , R be the points in which the mean axis of figure, the axis 
of the deformed figure and the instantaneous axis meet the plane C xy. Taking the 
radius of the spheroid as unity, the coordinates of P are 9. : , — 9 X , or 
2(f) sin X {t — t), — 2(f) cos X {t — r). 
The coordinates of P 7 are 0 9 6 ,, — ( 9 , — - —, or 
2 e + e 1 e + e' 
£ £ 
2( £ sin x - r )> ~ 2 ( t ) e + y cos x (t - r )’ 
• • 
and the coordinates of R are Oja), 6 2 /o), or 
o o 
— 2(f) 6 - sin \ (t — r), + 2(f) ~—- cos X (t — r). 
e + e ' e P e 
We thus see that the points P, P 7 , R all lie on the same straight line through C, 
and that this line revolves about C with uniform angular velocity X relatively to the 
moving axes Ox, C y. These axes are themselves rotating with angular velocity co 
about C in the same direction, and, hence, the actual angular velocity of the line 
PP'R, about C is <w {1 + e 3 /( e + e )}- The distances of R, P, P 7 from C remain constant, 
while since CR is of the order e compared with CP or CP 7 , which are themselves 
small quantities, we may suppose the point R to coincide with C. We conclude that 
