ME. J. H. JEANS ON THE STABILITY OF A SPHEKICAL NEBULA. 
41 
The Unsymmetrical Configurations of a Nebula. 
The Second Series of Equilibrium Configurations. 
§ 39. Let us now try to examine the second series of equilibrium configurations, 
which, as we have seen, is a series of stable configurations replacing the series of 
Jacobian ellipsoids. In this way we shall be able to gather some evidence with a 
view to forming a judgment whether the behaviour of the nebula after leaving the 
symmetrical configuration is such as is required by the nebular hypothesis. 
Let us suppose, in the first instance, that the symmetrical configuration from which 
this series starts is one in which there is no rotation, so that the configuration is one 
of perfect spherical symmetry. If the nebula is one in which cooling takes place 
very slowly, the configuration of the nebula will always be very approximately an 
equilibrium configuration. This configuration will be one of the spherically 
symmetrical series until the first point of bifurcation is reached ; after this the 
configuration will change so as to move along the other series, which passes through 
this point. 
Now we have already found the manner in which the configuration first diverges 
from spherical symmetry : in other words, we have a knowledge of the unsymmetrical 
series in the immediate neighbourhood of the point of bifurcation. If then, we can, 
by some method of continued approximation, obtain a more extended knowledge of 
this series of configurations, we shall be able to trace the motion of a nebula which 
is cooling with infinite slowness, and in this way form some idea of the motion to be 
expected in the more general case. 
Let us assume, as a general form for the “ series ” now under discussion, 
P — Po + Pl^l + p-p 2 + P3P3 +.. • 
where P s is the zonal harmonic of order s, and p 0 , p L , p. 2 are functions of r and of 
some parameter a. This parameter determines the position of any particular 
configuration in the series. We shall suppose that at the point of bifurcation a = 0, 
and we then know that when a is very small the limiting form of p is 
P — Po + 
In the notation which lias been in use throughout the paper, we find that 
corresponding to the density distribution given by equation (72) the gravitational 
potential at the point r , 9 is 
where 
V — $0 "U 6*1-Pl ~b 02^2 + ^ 3^3 ~b • • 
47T 
(116), 
9,. = 
2s + 1 l_r' !+1 J R 
1 p s r“ +z dr -f r’^ Ps '^ 
11—l 
VOL. CXC1X.—A. 
G 
(117). 
