180 
MR. J. H. JEANS ON THE CONFIGURATIONS 
and two similar equations, accented letters all referring to the new figure of 
equilibrium, and the value of 0' being now given by (cf. equation (41)) 
a/ _ W 
tt a'b'c’ 
so that, by comparison with equation (41), 
O'a'b'c' = 6abc .(102) 
There will, of course, be a solution for the compressible mass corresponding to each 
solution of these equations, which are virtually equations giving a solution of an 
incompressible mass under tidal forces. 
When the main tidal force disappears (tj = r 2 = t s = 0) the solutions will corre¬ 
spond, except for small tidal terms, to Maclaurin spheroids and Jacobian ellipsoids. 
The compressible solution which corresponds to the Maclaurin-Jacobian point of 
bifurcation will represent a point of bifurcation for the compressible mass. Let us 
proceed to determine this point of bifurcation exactly. 
At the point of bifurcation the configuration is spheroidal, so that e l = e 2 . By 
subtraction of corresponding sides of equation (100) and the similar equation in b, 
we obtain 
or 
{a ,2 -b' 2 ) (J' AB -6'la' 2 b' 2 ) = 0.(103) 
The spheroidal series is determined by the vanishing of the first factor, and the 
ellipsoidal series by the vanishing of the second factor. At the point of bifurcation 
both factors vanish, so that a' = b', and 
( 101 ) 
O', 
6ct 2 c .(104) 
This equation, together with (100) and its companion, will determine the values 
of a', d and w at the point of bifurcation. 
26. The values of these quantities differ by terms of the order of e from the 
corresponding quantities a , b, c at the Maclaurin-Jacobian point of bifurcation, 
so that we may suppose that 
~2 — —2 4" 6 A ! — 2 ), &C., ft/ 2 — w 2 + e Aw". . . . . (105) 
Cl Cl \Cl J 
or again, using relation (10: 
«' 4 J' AA - 
«VJ' AA = 
The equation of the figure of equilibrium is now 
^±^ + yl +e 
2 T 7 2 Te 
a b 
{x 2 + y 2 ) A 
a‘ 
+ z 2 A 
+ 
La * 4 
er 
+ 
