OF ROTATING! COMPRESSIBLE MASSES. 
169 
value of Q so obtained to the value given by equation (34), the general equation of 
equilibrium is obtained in the form 
po 
[V,■ (1) — eE,-] + (t x x 2 + r 2 y 2 + t 3 z 2 ) + (x 2 + y 2 ) 
= cons. — 7 -p 0 abc6 
25+ t P„)-M 7 -2)(2:^ + ,P„ 
a 
X 
a 
+ ie S ( y -2)(y-3)(2±- 2+ eP l 
.X 2 
a 
(44) 
12 . The first approximation has been obtained by putting e = 0 in this equation; 
we now proceed to higher approximations. A second approximation will be obtained 
by omitting all power of e beyond the first; a third approximation by not going 
beyond e 2 , and so on. 
On replacing eP„ by eP 0 + e 2 Q 0 + e s R 0 +-..., we may suppose that the density expanded 
in powers of e is 
so that the boundary, p = cr, is 
2 — +eP 0 + e 2 Q 0 + e 3 P 0 + ... = 1 .(46) 
cr 
The general equation of equilibrium will be obtained from equation (44) on 
replacing eP L) by 
fPo + e 2 Q 0 + e 3 Ro +.(47) 
The value of V, (l) to be used in equation (42) will no longer be that given by 
equation (36); let us suppose the whole value to be 
V I (l) + eAV t (l) + eW ; (l)+...,.(48) 
this being the internal potential of a homogeneous solid of unit density whose 
boundary is determined by equation (46). Similarly, let the whole value of E, be 
supposed expanded in the form 
E ) + eAE i + e 2 ciE i +.... 
Finally, let the value of « 2 in the complete solution be supposed given by 
27 -p^cibc 
tl + e An + e“ Sn +... 
(49) 
