MR. S. S. HOUGH OH THE ROTATION OP AH ELASTIC SPHEROID. 
335 
and from (18), (26) 
£ — £o + £] — 
2e 2 
f 2 xz — 9 x yz) + 
a (1 + e'/e) 
1 + I - 9 ^), 
--- (te - 0 x yz) 
T {fTT/fl 
• P9). 
6 . Determination of the Period. 
The method we have followed hitherto has enabled us to express the displacements 
at any point of the body by means of two arbitrary constants 0 X , 0. 2 , but the 
quantity A, whose value it is our chief object to determine, has entirely disappeared. 
We have, in fact, verified that the equations (7), (8), and the boundary conditions (9) 
are approximately satisfied by the forms (28), while to the same order zero is the 
approximate value of A. We require now to have recourse to a method which will 
enable us to carry our approximations to the value of A further. 
By using the well-known equations of motion of a body of changing form Professor 
Woodward* has shown that the determination of the period may be reduced to the 
evaluation of the disturbing angular velocities due to the flow of the material 
relatively to the principal axes of the body. As we have now expressed the 
displacements, and consequently the velocities, at any point in terms of 0 X , 0 2 , 
which may be taken as the displacements of the body as a whole, we are in a 
position to calculate these disturbing angular velocities. We might then introduce 
their values in Professor Woodward’s equations and proceed to the determination 
of the period by his method. We propose, however, to make use of equivalent 
equations which express that the rates of change of angular momentum for the 
system as a whole about the axes Ox, O y are zero. It seems somewhat preferable, 
on account of the additional simplicity of the motion of the axes themselves, to 
refer to these axes rather than to the moving axes used by Woodward, which 
correspond with our axes Oaq, O y x . 
If h x , h 2 denote the components of angular momentum about Ox, O y, we have 
- ( v + 
xoi) zj dm, 
where dm denotes an element of mass, and the integral is taken throughout the 
volume contained by the displaced surface. Peplacing the integral by an integral 
taken throughout the mean volume and a surface integral, we have 
h x 
|| {(ivy — vz) — cox zj dm — p 
£coxzd S, 
* ‘Astronomical Journal,’ xv., Ho. 34-5. 
