MR. S. S. HOUGH ON THE ROTATION OF AN ELASTIC SPHEROID. 
331 
, 0 
xJjx + n (r ^ - 1 j u 1 + n~ (u L x + vyy + ivf) - {vy-cjQ 
x 
= x . or (9»xz — 0 } yz), 
/ 3 \ 3 
Dj + » (»’ 07. - 1J Vi + n ^ {u x x + v L y + w x z) - « - g£ x ) y 
— y. co z (O z xz - 0 x yz), 
fa + ?l ( r b - w \ + 71 b K* + v iV + w i z ) — z 
dz 
= z . or (Oyxz — 9 1 yz). 
J 
( 20 )* 
To the same order we may suppose these equations to hold good at the surface r = a, 
instead of at the surface r = a {1 -f- eT 3 }. 
§ 5. Determination of the Elastic Distortions. 
The modified forms (19) of the equations of motion are simply the equations to 
which we are led in determining the displacements in a strained elastic solid and the 
solutions of them, when either the displacements or the surface-tractions at the 
surface of a sphere are given, are well known.t We may readily adapt these 
solutions so as to satisfy the boundary conditions (20). 
Denote for brevity the function w 2 (9. 2 xz — Oyjz) by S 2 . 
From (19) we obtain at once V 2 t// = 0. This equation replaces the more com¬ 
plicated form (8); a particular solution of it is i// = AS 3 . 
Introducing this value of ip into the left-hand members of (19), we find 
V\ = 
a as, 
n dx 
A oS 2 
n 3 z 
particular solutions of which are 
u 
l — 
1 A ,3 
1 0 n dx ’ 
_1_ A .3 ^§2 
10 n ' dy 
u\ 
x a 2 Bsq 
10 n dz 
These do not satisfy the last of equations (19), but they make 
du 1 .dv 1 dw J __ 1 A 
dx ^ dy ^ dz ~ 5 n ~ 
* For the physical significance of these equations, vide § 8 infra. 
f Vide Love, ‘ Elasticity,’ vol. 1, chap. 10. 
