32 
ME. J. H. JEANS ON THE STABILITY OF A SPHEEICAL NEBULA. 
some spherical surface harmonic S„ of order n. The frequency of vibration is 
independent of the particular spherical harmonic chosen, depending only upon the 
order n. 
The vibrations of order n — 0 have been seen to be of no importance ; the stability 
of the vibrations of orders different from zero has been discussed, in the limiting case 
in which the nebula extends to infinity, with the following results :— 
Starting from any stable configuration of spherical symmetry, the vibrations of any 
order n, different from zero, all remain stable until the function m„, defined by 
equation (104), passes through a certain critical value. In any case this critical 
value is first attained for a vibration of order n = 1. 
For a nebula which actually extends to infinity, the critical value is u x = 1. 
When this value is reached we come to a second series of equilibrium configurations, 
the form of which will be investigated later. If this value is passed, the configura¬ 
tion remaining spherical, there will not be vibrations in which the time enters 
through a real exponential factor, because the critical vibrations remain of frequency 
p — 0, the inertia of the nebula being infinite. 
If the radius It 1 of the nebula is regarded as very great but not infinite, this 
statement is not true, since the inertia cannot now become infinite. In this case the 
first new series of equilibrium configurations is again reached when (n) r=I?i attains a 
certain critical value, and the critical vibration is again of order n — 1. The critical 
value of (w),. =El has not been calculated, but when becomes infinite, it has a 
limiting value which has been shown to lie between I and 1-g-. 
Taking y — 1, we have as the value of u x , 
n - L' - pr ~ 
( 106 ). 
The question of stability turns entirely upon the value of this function, which may 
appropriately be termed the “ stability-function.” 
We now see that the whole question of stability depends upon the ratio of the 
density to the elasticity at infinity. This result is not hard to understand. In the 
first place, since the nebula extends to infinity, we may, so to speak, measure it upon 
any linear scale we like. If we measure it on a sufficiently great scale, the nebula 
still remains of infinite extent, but the variations in temperature or structure which 
occur near the centre can be made to appear as small as we wish, and the solid core 
can be made to appear as insignificant as we wish. Thus by measuring any nebula 
upon a sufficiently great scale we can make it appear indistinguishable from an 
isothermal nebula, and the critical vibration for which 'p = 0 does not disappear from 
sight, since in the limit this vibration (measured by £/r) remains finite at infinity. 
Further, as Professor Darwin points out, we can make it appear like a nebula in 
