MR. S. S. HOUGH OH THE ROTATION OF AN ELASTIC SPHEROID. 
329 
If now we change our variables from u, v, w to u x , v v w x> the equations of motion 
become 
nV z u x + \~ll x -f- 2o)i\v } — — dxp/dx — \' 2 z# 3 + 2 oj i\z6 l 
nV 2 v 1 -j- Xhq — 2 <ai\u x — — dxft/dy + X 2 z6 x + 2 a>i\z6. 2 ' 
}* • • (1 • 1 )’ 
n'V 2 w l + \ 2 w 1 = — dxp/dz -f- (a?# 3 — yO x ) 
dufdx -f 0iq fy -j- dwjdz =0 J 
while in virtue of (14) the boundary equations may be written 
du. 
I U..J du x difj 0//, dv 1 diffj 
ib cos a + ^ cos a + V cos p + ^ cos y + — cos a + v cos p 4- cos y 
r '' Ox Oy cz ‘ Ox ox ' 0X / 
So 
dy‘ ‘ oz ' ’ Ox 
( v\ — gh) cos a. — or (6gxz — dyyz) cos a, 
0y, 
0V 
du r 
di\ 
du\ 
cos 
y 
\ji cos /3 + n j y ‘ cos a + ^ cos /3 -J- 1 cos y + yv cos a + 1 cos /3 
— (-y'j — < 7 ^) cos f — or (fyxz — f^yz) cos /3, 
[ dio l 0Wj 0 . 0O 0/q . 0c, „ 0w, 
if/ COS y + n | 0 COS a -f- , COS P + y, COS y -r 07 COS a + yy COS P + -yy cos y 
— — (/Ci) cos y = or (0. 2 xz — 0 } yz) cos y. 
§ 4. Reduction of the Equations when the Body, is of the form of a Spheroid of 
Small Ellipticity. 
The only assumption we have made as yet as to the form of the free surface is that 
it is a possible figure of equilibrium for a rotating mass of liquid, so that the body 
may be free from strain when rotating uniformly. The simplest form which can 
occur and that which presents the greatest interest is the case of a spheroid of 
revolution of small ellipticity e. We propose for the future to confine ourselves 
to this case. If we neglect the square of e the angular velocity of rotation is related 
to e by the equation 
e = loor, 167rp.(17), 
where the density p is expressed in gravitational units. If then the units of length 
and time be so chosen that p is finite, o/ will be a small quantity of the order eh 
Take as the equation to the free surface r = a (1 + «T S } where 
T 2 = ( x 3 + y 2 — 2z~)/3a 2 . 
2 u 
MDCCCXCVI.—A. 
