OF ROTATING COMPRESSIBLE MASSES. 
181 
and this agrees exactly with the figure previously considered if we take 
A 
\a J a 
Using relations (105) and (106), equation (100) transforms to 
O' 
(106) 
abc 
J 
00 
A 7rp tj abc 21rp 0 abc a 2 . 
+ eabc 
4ej + p ( 2 J AA 2 I AA ) — 2 I AC — 
CL 
V 
? 
A 
00 
_ 
2 irp 0 abc a 4 _ 
0 . . 
(107) 
The first line vanishes in virtue of equation (38) and the equation reduces t< 
4e, +p ( 2 J AA - 4 O - - 2 Iac - = An > 
' a ! c a 
. . . (108) 
which agrees with equation (86), as of course it ought to do, and there is a similar- 
equation agreeing with equation (88). 
27. Equation (104) may be written in the slightly symbolical form 
a 6 cJ AA + e A (a 6 cJ AA ) = 0a 2 c, . (109) 
and the condition which determines the ordinary Maclaurin-Jacobian point of 
bifurcation is 
« 4 Jaa = 0 . 
Thus equation (109), which expresses the condition to be satisfied at the point 
of bifurcation on the compressible series, reduces to 
A(a*cJ AA ) = 0. 
( 110 ) 
Now from the definition of we have 
a 6 cJ AA 
■“_ dX 
0 ( I +x/a 2 ) 3 (1 +\/c 2 )- ’ 
whence we readily find that equation (110) is equivalent to 
6I A ‘(|) +Im (?) = # ' • 
(m) 
Equation (ill), together with equation (108) and its companion, determine the 
values of p, r and A n at the point of bifurcation on the series of compressible 
configurations. 
VOL. ccxviii.—a. 2 B 
