ME. J. II. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
BO 
Exchange of Stabilities. 
§ 28. We have now completed an investigation of the configurations at which 
a transition from stability to instability can occur, as regards the spherical form, for 
vibrations of orders different from zero. It is unnecessary to discuss vibrations of 
order n — 0, for the following reason. 
Our problem is to determine the changes in the configuration of a nebula which 
will take place as the nebula cools, starting from a spherical configuration, supposed 
stable. We are not concerned with the succession of spherical configurations, but 
only with an investigation of the conditions under which a spherical configuration 
becomes a physical impossibility. Now a point of bifurcation of order n = 0 does 
not indicate a departure from the spherical configuration. It indicates a choice of 
two paths, one stable and the other unstable, and the configurations on both paths 
will remain spherically symmetrical. 
We have therefore determined already the circumstances under which a transition 
from a symmetrical to an unsymmetrical configuration can occur. It remains to show 
that there is, in effect, an exchange of stabilities at a point of bifurcation, and to 
examine on which side of the point of bifurcation the spherical configuration is stable. 
We are going to prove that the spherical configuration is stable for all values of u 
less than u 0 , the lowest value of u at which a point of bifurcation of order different 
from zero can occur. Our method will be as follows : Any two equilibrium configura¬ 
tions can be connected by a continuous linear series of equilibrium configurations, and 
u will vary continuously as we move along this series. If one of the two terminal 
configurations is stable, and if the linear series can be chosen so that u does not at 
any point of it pass through a value for which a vibration of frequency p = 0 is 
possible, then we know that the other terminal configuration is also stable. 
The value of y, the gravitation constant, has been taken equal to unity. If this 
constant is restored, the value of u becomes (equation (54)) 
u — 
277 ypE 
dp j dm 
dr / dr 
Since cr = XTp, we have 
dm 
dr 
= p A (XT) + XT A 
dr 
For an infinite nebula, the first term on the right-hand of this equation will 
vanish at infinity in comparison with the second. Hence we have as the 
value of u x 
0I _ Lf, 2ir 1P r ~ 
r =go 
(104). 
