80 
MPv. J. H. JEANS ON THE EQUILIBRIUM 
hence equation (50), at j^oints inside both circles, becomes 
2Trpl 
0 npfr I 1 
CO- 
-{-rrp t (cQ, a., — a, (1 — + i'-) - a constant. . (51). 
j! = l 
We must therefore have, for all positive integral values of n, 
0 
lirpi 
f, («i. .■■•) = oj 1 - 
(52). 
The condition that equation (46) or (47) may represent an equilibrium configuration 
is therefore 
/i/«i =/y'Q=/3/«.3 = ..(53), 
and the value of w" is such that each fraction is equal to 
Ct)^ 
lirp 
Since for anv 
equilibrium configuration the axis of rotation must coincide Avith the centre of gravity, 
we see that if the coeflicients of (46) satisfy (53), the curve must he referred to its 
centre of gravity as origin. 
The points of bifurcation are given by the Hessian of this system of equations. 
For our purpose this may be most conveniently written in the form 
^/i (l/j 3/s 
1 — 
iTVpj 
ca. 
va. 
3/i 
f- - b - 
0«., \ 
?f 
diu ’ 
• 
( O ' 
iirp' 
= 0 
(54). 
Uur method will, for reasons already explained, break down as soon as a cusp or 
branch point occurs on any linear series. Any solution will be a true solution 
provided we can pass from this solution to another, known to be a true solution, over 
a path througli a system of linear series, Avithout passing through a point at Avhich 
a cusp or branch j^oint occurs. Now the occurrence of a cusp or branch point indicates, 
in the physical joroblem, the division of the mass of fluid into tAvo separate masses, 
and Avhen tliis occurs the solution breaks doAvn for a second reason also : for 
equation (49) is only true AA’hen the surface of the fluid is continuous; Avhen the 
surface consists of tAA'o distinct parts, the constant on the right-hand side has difterent 
A’alues for the tAvo parts of the surface. 
This limitation Avill not cause trouble in the present inA-estlgation. For Ave are 
only desirous of tracing the changes in the configuration of the fluid up to the 
separation into tAvo parts, and eAmn if our method l\ad enabled us to proceed beyond 
this point, it Avould have been fruitless to do so. 
