328 
MR. S. S. HOUGH OH THE ROTATION OF AN ELASTIC SPHEROID. 
x , y, z due to the attraction of the body when it is rotated without distortion through 
small angles 0 V 6„. If then x 1} y l5 z 1 denote the coordinates of this same point referred 
to axes obtained by rotating the old axes with the body, on putting in evidence the 
arguments of the function Y 0 , we have 
V 0 {x, y, z) + v' 0 = V 0 (x 1} y lt zj) .(13). 
But the direction cosines of the two sets of axes are given by the scheme 
X 
y 
z 
x l 
1 
0 
~ 02 
Vi 
0 
1 
0i 
z i 
0-2 
-*1 
i 
whence x x = x — z0 2 , y 1 = y + z9 v z 1 = s + x0 2 — yO x . 
Introducing these values in (12) and expanding by Taylor’s theorem, we find 
V 0 (x, y, z ) + v' 0 = Y 0 (x — z0 2 , y + z0 lt z + x6, — y0 x ) 
av, 
avn 
= V 0 {x, y, z) - z0 2 ^ + Z0 1 + (x0 2 - y0 1 ) 
BY 
& 
But from the definition of g we have, at the surface, 
av, 
a y 
av 0 o Q av 0 av 0 
g COS a — — - to-x, g COS /3 = - ~ 0 J~y, g COS y = - -yy 
and, therefore, 
v' 0 = z0 2 (g cos a + od 2 x) — z$ l (g cos /3 + io~y) — (x0 2 — y0 1 ) g cos y 
= g (v o cos a + v 0 cos /3 + iv 0 cos y) + ad ( 0 2 xz — O^z), 
or 
whence, finally, 
v'o - gt o = oj z (0 2 xz - 0gjz), 
v ~gC = (o z (0 2 xz — 0gyz) + v\ - gh . . . 
(14) 
