194 
MR. J. H. JEANS ON THE CONFIGURATIONS 
Multiplying by 2, 1 and adding, 
Sn = -2d(^- + -\ ■ .(158) 
\ a cl 
an equation which may be used to replace (156). 
By direct computation from equation (136), I find 
A 3 = 0T9769u' —0'58215v' 
+ 0-06706r / T0'01164s / -0-61497t / 
+ 0-04270R , + 0 03778S / —0-07530T'—1-1894U' 
= 0T9769ifi —0‘58215v / + 0’02348 — O'l 1113 (y-2), 
so that equation (157) reduces to 
0’39538u / + 2’8142v / = 0’0212 + 0’0794 (y -2) .(159) 
Equations (158) and (159) are adequate to give if and v' in terms of any assigned 
value of Sn. We require especially the solution at the actual point of bifurcation of 
the compressible series, and to obtain this the two equations must be combined 
with a third equation expressing the condition for a point of bifurcation. 
The third equation is 
$ (a 6 cJ AA ) = 0, 
or, transforming by the method of § 27, 
6u ' T . y/ T _ 24 i^ TT i 6 P r TT . 37 ’ 2 TT 
2 -*-AAA ' 2 ^AAC 4 -^*-A 4 “T 2 2 ^A S C ' 0 4 J^A^C 2 - 
a c a arc 2 c 
Inserting numerical values, this becomes 
0'24919u' + 0‘20878v / = CT00024.(160) 
From this and equation (159), I found by direct solution 
if = —0'00550 —0‘02651 (y-2) 
v' = 0‘0U778 +0'03195 (y — 2)_ 
and equation (158) now gives 
(161) 
Sn =—0-01292 —0-05495 (y-2).(162) 
The solution which has just been completed, combined with the two exact 
solutions (127) and (128) previously known, will give the exact solution for a 
compressible mass at the point of bifurcation at which the pseudo-spheroidal series 
gives up its stability to the pseudo-ellipsoidal series. 
