332 
MR. S. S. HOUGH OH THE ROTATION OF AH ELASTIC SPHEROID. 
We must therefore add to the particular integrals just found complementary 
functions which satisfy the equations 
V 2, tq = 0, Vhq = 0, Vhiq 
3 o x 3 U) x _ 2 A Q 
da + a,. + A.. — + 5 
= 0 
3 y 
( 22 ). 
Now the functions r 7 ( ~f), r 7 ~ (~j\ r 7 ^ (~r) are spherical harmonic func- 
u \ r 
3y \r 5 /’ 3 z \?’ 5 
tions which remain finite at the origin, and hence, if we take 
u, = 
v, = 
w, = 
B f* + O 7 1 (b 
dx dx V r° 
B f+ C ’ J !(T t 
By-(- O 7 1 
dc d-2 \ 7' 
(23) 
the first three of equations (22) will be satisfied identically, and the fourth will also 
be satisfied if 
3.7.0 = f —, 
J n 
or C = 
O 
TO 5" 
A 
n 
Therefore, adding together the particular integrals (21) and. the complementary 
functions (23), we obtain as a set of solutions of the equations of motion which remain 
finite at the origin 
* = 
= AS, 
\9 
A 
o 38, 
B 
3S 2 
A 
u x 
1 
1 0 
n 
— w 
dx 
+ 
dx 
" To A 
— r 
n 
A 
o 3S, 
+ 
B 
3S, 
A 
V 1 
—■ 
1 
1 0 
n 
Y* —- 
« 
3y 
To ¥ 
- r 
n 
A 
, 3S, 
+ 
B 
3S 2 
A , 
w l 
= ■ 
1 
1 0~ 
n 
dz 
3? 
_ TOT 
% 
.7 
dx \ r° 
a / c< 
3 / Ss 
dz \ r 5 
1 
• (2-t). 
j 
We must now prove that these solutions are sufficiently general to satisfy the 
boundary conditions (20) at the surface of the sphere r = a. 
From (24) we obtain 
«,* + v, V + V h z = - i ± 1 * S 2 + 2BS, + * f) "S. = -iz + 2BS* . (25). 
7 1/ Jv 
n 
