OF ROTATING COMPRESSIBLE MASSES. 
1(33 
8. Suppose that P„ is expressed in the form 
p = y (— y~ 
u W’ ir c 2 . 
and let P be a function of x, y, z and a new variable a, given by 
p = fM® 
q 2 y 
q 2 z x 
q 2 a 2 + /ul ’ q 2 h 2 + fx ’ q 2 c 2 + fx 
so that P„ is the value of P when p. = 0. Further introduce >/, defined bj r 
so that 
f = x/(q 2 a 2 + fx), &c., 
p = F(^, ? v, g 2 n 
and f and D (an operator) defined by 
f = 
+ —T’-1-~-1 — + p) ^ + • • • — 1 > 
gV + p q 2 h 2 + fx q 2 c 2 + p 
D = 
qW ^ q 2 a 2 + mJ 0f 2 + 1 gW g 2 6 2 + p/ d,/ 2 ' VgV gV + p/ 
( 1 
i \ a 
+ 
j__ 1 \ a 8 
I e/2 
Let </> (g), a function of >/, f', p and g, be given by 
<p(g) = 
p - i/ DP+ sKi/TD*- ^(j/f dt+ 
— — e 
DP 2 - - /D 2 P 2 + — / 2 D 3 P 2 -— /‘ 3 D 4 P 2 + 
8 J 192* 7 9216 
— e 3 [D 2 P 3 -...], &C. 
192 L 
( 12 ) 
When p = 0, D = 0 and P = P 0 , so that 0 (g) reduces to eP 0 , and 
^ 0 (?) _ 1 {^L , 2 /! 
" g 2 la 2 b 2 c- 
q 
./ + — ~i2 ( 72 + 72 + — + ePu ~ ? 2 ) • 
Thus the surface of constant density p is the surface p = 0 in the family of surfaces 
/+0 (g)/g 2 — o.(i ; 3) 
