330 MR. S. S. HOUGH ON THE ROTATION OF AN ELASTIC SPHEROID. 
Then the direction cosines of the normal to the surface r = a {1 + eQ„}, where Q /( 
is a solid harmonic of order n, are 
i nx „ 3Q„ i „ 
7 + ae - g7 h &c -> 
and thus we have 
whence 
or 
« , J 2a? 0T 2 ] 
cos a =- \- ae\— To — -7 y , 
r [ r- ~ dx j 
- » i_ J 2 V m 
- 7 
cos (3 
cos y 
* I 
— 1“ Ct€ 
I? 1 
3T, 
3s 
£o — z ^2 cos a ~ cos — ( 9 2 x — 6 j?/) cos y 
= ae 
= ae 
, ( 3T 2 
X-tt 
0T, 
*£)+'■ 
0^ 
y 
BTo\ 
n 2xz , a 2 v z 
Co = - 7 [%** ~ ^1^] 
( 18 ). 
Now observation indicates that in the case of the Earth the oscillation in question 
differs but slightly in type from the motion of a rigid body, whence we conclude that 
u Y , v x , u\ must be small compared with u, v, w. We propose therefore to make the 
assumptions, leaving the verification thereof to our subsequent work, that u 1} v lt 10 v xp 
contain the small factor ad, while X is a small quantity of the order ad. The latter 
assumption is justifiable when the body is perfectly rigid, since in this case we have 
A _ Of - & _ _ load 
a> 167T/3 
If then, we neglect small quantities of the order ad or e 3 , the equations of motion 
(15) reduce to 
B-vk 
n\- J u 1 = — yy > nV z v 1 = 
dxjr 
dy 
> nV' 2 w 1 = 
3\fr 
0^ 
1 1 _ n 
0 .T By Bs 
(19). 
Since all the terms in the boundary equations involve the small factor ad, we may 
replace cos a, cos / 3 , cosy, by xjr, y/r, zjr respectively, with errors only of the same 
order ad, and thus the approximate form of the boundary equations is 
