OF ROTATING COMPRESSIBLE MASSES. 
183 
The value of D/q 2 is independent of q, and <j> (q) may be rearranged in the form 
*(?) = « [P-i (g 2 /) ^ P+* (g 2 /) 2 (fff P 
whence, on differentiation 
so that 
90 (?) = 
dq 2 
L i P P-A (g 2 /) 
2 [ l — C 0 (?) ) - g 2 
0^ 
1-7T P + ' 
4 g 2 
a? 
32 \ ar + A/\? 
-.Yp 
-3V‘<|) 2r 
This gives on integration, since D/q 2 is independent of q, 
e D . e 
2 
X 
D 
1-— - P+ 
4 q 2 32 \ a 2 +\l\q 2 ,, 
. 
( 121 ) 
The value of q 2 on the right of this equation is the root of equation (117), <p{q) 
being given by equation (118) in which f is in turn given by equation (119). Thus 
q 2 will be obtained by the elimination of <p (q) and f from equations (117) to (119), 
omitting terms in e 2 . From these three equations we obtain 
gV=-i> (a) = -* [P—i/DP+*/W], 
so that f is a small quantity of the order of e, and, omitting terms in e 2 , 
0 (?) = eP, 
which is independent of q. Equation (l 17) now gives, as the value of q 2 , 
q 2 = 2 
x 
er +A 
+ eP. 
The coefficient of e on the right hand of equation (121) now reduces to 
— P2 
x 
a 2 + \ 
1-2 
x 
2 \ 2 
a 2 + \ 
iD 
x 
2 2 .. 2 
q \ a 2 + \J\q~ 
D 
i 
96 
1-2 
a; 2 
a + A/ J \q 
DV 
P. 
( 122 ) 
The value of P is given by equation (56), and that of D/q 2 by equation (120). 
From these we obtain 
^ P = 2 (\ - J—) (12L£ 2 — int] 2 + 4:m^ 2 + ip), 
q 
T) 2 
-JP = 2 
?‘ 
a a+ A 
24L (— -— Y — tt 1 
' a- a + A 1 
A. 
1 Y1 
Vc 2 
c 2 +A/_ 
2 B 2 
