OF ROTATING COMPRESSIBLE MASSES. 
I 67 
The equations of equilibrium (l ) now assume the form 
V-2<V 
Q 
K 7P 
ox 
-L- — & c 
OX 
and these have the common integral 
diZ-V-i = Q + 
— 1 
cons. 
(30) 
We have already (equations (3) and (5)) assumed the value of p to 
be 
P = Po 
l_ e2 ^_ e »P 0 
a 
(31) 
and on expanding p y 1 by the binomial theorem, equation (30) becomes 
Q = cons. + Kyp y ~ l 
-(e2^+e 2 P 0 )+l( 7 -2)(eZ^+e 2 P 0 
a 
, x 
a 
—Hy—2) (y—3)f e2^+e 2 P 0 ) + ... 
\ W 
(32) 
Let the figure of density given by equation (31) be supposed to be a figure of 
equilibrium under a rotation w and tidal forces of potential V T where 
V T = t x x 2 + T 2 y 2 + r s z 2 , 
. . (33) 
in which of course T l +T s + r :i = 0. We then have at any point inside the mass 
a = v.+Vj+K (»*+»*) 
= Po [V,- (1) - eEf] + (t,x 2 + T 2 y 2 + -f) + ho 2 (.X 2 + if), . . . . (34) 
and on equating the right-hand members of equations (32) and (34) we obtain a 
solution of the problem. 
11. The simplest solution occurs when the variations of density are small, so that 
e is small; in this case /cy is large. We must not simply put e = 0 in equation (32), 
for otherwise Q would reduce to a pure constant. We accordingly suppose xye 
to remain finite, and put e = 0 in all remaining terms. Thus the equation to be 
solved is 
poVi (1) + (t,X 2 + r- 2 lf + T;,Z 2 ) + W {pc 2 + if) = cons. - xyep, 
Y-l 
i X 
2 
a 
+ V- + - 
+ b 2 + c 2 
(35) 
