202 
MR. J. H. JEANS ON THE CONFIGURATIONS 
Thus at the ellipsoidal point of bifurcation, the innermost strata of equal density 
are of flatter shape than those for an incompressible mass, showing that compressibility 
tends to postpone instability for the spheroidal mass. 
The value of w, the angular velocity at this point of bifurcation has been found to 
be given by 
w 2 /2tt/ 3 0 = 0'18712 —0'04400e —(0'01292 + 0'05495 (y —2)) e 2 —. . (177) 
or, if we evaluate o> 2 in terms of p, the mean density of the whole mass, 
a) 2 /27rp = 0’18712 + 0'06827e + (0‘01602 + 0'07098 (y — 2))e 2 —. . (178) 
the coefficient of e 2 now being only approximate. We notice that « 2 /2x^ is greater 
for a compressible mass than for an incompressible mass, so that again compressibility 
may be said to postpone the instability of the spheroidal form. # 
Equation (176) applies only to the innermost strata for which x i , &c., may be 
neglected. The equation of the outer strata, both at the point of bifurcation and 
elsewhere, are found to contain terms of degrees four and higher, there being terms of 
degrees four and two multiplied by e, terms of degrees six, four and two multiplied 
by e 2 , and so on. The terms of degrees six and four have been calculated for the 
ellipsoidal point of bifurcation and the terms of degree four for the pear-shaped 
point of bifurcation. 
The presence of these terms destroys the spheroidal or ellipsoidal shape of the 
outer strata. In general it is found that the outer strata are more lens-shaped than 
the inner strata when these latter are spheroidal, and more spindle-shaped than the 
inner strata when these latter are ellipsoidal. The lens-shaped form of the outer 
strata may go so far that the outer boundary develops a sharp edge. When this 
occurs, centrifugal force is exactly equal to gravity at points on the periphery of the 
lens, and any further increase in the rotation of the mass results in matter being 
thrown off from round this periphery. Similarly the spindle-shaped figure may 
develop sharp ends, in which case matter will be thrown off here also. 
47. Consider now a gradually shrinking mass of gas or other compressible matter, 
the rotation increasing as the shrinkage proceeds. For a very slow rotation the 
strata and the boundary will all be spheroidal. As the rotation increases, the 
boundary departs more and more from the spheroidal form, taking a series of forms 
* All terms in w 2 /27rp are positive because y-2 is necessarily positive; for, as we shall see, if y<2, the 
compressibility so far postpones the occurrence of the ellipsoidal point of bifurcation that it does not occur 
at all. Incidentally equation (178) has an important bearing on the origin of the solar system. It shows 
that for every mass which has broken up by fission <u 2 /2irp must at some time have exceeded the value 
0T8712. This provides the keystone, which has so far been wanting, in an argument I have given 
elsewhere (‘M. N. Royal Ast. Soc.,’ 77, p 191) to show that the solar system is very unlikely to have 
broken up by rotation alone. 
