OF ROTATING COMPRESSIBLE MASSES. 
203 
which we have called pseudo-spheroidal, these being more lens-shaped than true 
spheroids. There are two alternatives. It may be that the point of bifurcation will 
be reached before a sharp edge forms at the equator of the pseudo-spheroids, and 
if this happens the innermost strata will become ellipsoidal, and the outer strata 
and the boundary will be pseudo-ellipsoidal; the mass will proceed towards the pear- 
shaped form and ultimate fission. But, instead of this happening, it may be that a 
sharp edge will be formed before the point of bifurcation is reached, and the mass 
will disintegrate through equatorial loss of matter. It is of the utmost importance 
to determine which of these events will happen first for a particular mass. 
48. Some information may be obtained from a general survey of the problem. 
The spherical solutions for a mass of matter at rest obeying the law p = Kp y have 
been investigated by Bitter, Darwin, Emden # and others. Excepting the special 
case of y = 2, in which the equation reduces to a linear equation, the solution can be 
expressed in finite terms in one case only, namely y = lit In this case the solution, 
first given by Schuster,! is 
P = A(l+r*/a 2 )-\ . (179) 
This happens also to give the lowest value of y for which the mass is finite. For 
values of y less than Tit the matter extends to infinity and the total mass is infinite; 
when y — 1 it, the matter extends to infinity but the total mass is finite; when 
y > lit, the matter is of finite extent and of finite total mass. As y increases the 
variations in density becomes less rapid and finally the value y = go corresponds to 
an incompressible mass in which the density is uniform throughout. 
Now clearly a mass for which y = lit will lose matter equatorially with even the 
slightest amount of rotation. For all except an infinitesimal fraction of the whole 
mass is concentrated in regions near the centre, and the potential near the edge may 
accordingly be taken to be M/r. The value of 13 is therefore 
13 = — + |<o 2 (x 2 + y 2 ). 
r 
The problem is now seen to be identical with one which has been studied by 
Roche.| There is a critical equipotential on which double points occur all round the 
equator, and this is the equipotential 
13 = f (Mai) 5 . 
* R. Emden, ‘ Gaskugeln ’ (Leipzig, 1907), where references to the work of previous investigators 
will be found. 
t ‘ British Assoc. Report,’ 1883, p. 428. 
I “ Essai sur la constitution et l’origine du Systeme Solaire,” ‘ Acad, de Montpellier. Section des 
Sciences,’ VIII., p. 235. A more accessible account is to be found in Poincare’s ‘Lemons sur les 
Hypotheses Cosmogoniques,’ 2nd edition, p. 15. 
