22 
MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
boundary, will, in every configuration of the gas, satisfy our equations with r p = 0. 
Such a motion, in fact, simply leads to a configuration which is physically 
indistinguishable from the initial configuration, and in which the potential energy 
remains unaltered. The motion which we have found from our equations is the sum 
of a motion of this kind, and a true limiting vibration. It is impossible to separate 
the two motions, except by considering vibrations of frequency different from zero, 
but fortunately the question is not one of any importance. 
§ 21. Let us now attempt to form the final equation in some cases of interest. The 
equations of an isothermal nebula at rest under its own gravitation have been 
discussed by Professor Darwin. # Our function u (equation 54) is given, in the case 
in which the nebula is isothermal, by the equation 
u = 
27rpr 2 
XT ‘ 
(78). 
and it will be seen that this is the same as the u of Professor Darwin’s paper. It 
appears that in general u cannot be expressed as a function of r in finite terms, but a 
table of numerical values of u is given, t The value of u approximates asymptotically 
to unity at infinity, so that at infinity p varies as r~ 2 . Darwin’s nebula extends 
from r — 0 to r = oo , but it is obvious that we may, without disturbing the 
equilibrium, replace that part of the nebula which extends from r = 0 to r = R 0 by 
a solid core of mass equal to that of the gas which it replaces. We may also remove 
that part of die nebula which extends from r — R 1 to r — °° , if we suppose a pressure 
to act upon the surface r = P : of amount equal to the pressure of the gas at this 
surface. We may suppose the medium outside this surface to be of any kind we 
please, but as it lias already been pointed out that the pressure can, in nature, only be 
maintained by the impact of matter, we shall suppose that this matter is of a density cr 
which is continuous with the density p of the nebula at the surface of separation. 
We may now write equation (52) in the simple form 
(79). 
We have, up to the present, supposed the nebula.to be acted upon by a spherically 
symmetrical system of forces in addition to its own gravitation. Now it is essential 
to the plan of our investigation that we shall be able to make the configuration of the 
nebula vary in some continuous manner, and this compels us to retain this generali¬ 
sation. We shall, however, suppose that when the nebula extends to infinity, 
u retains some definite limiting value u x , thus including the free nebula as a 
sjiecial case. 
* C4. II. Darwin, ‘Phil. Trans.,’ A, vol. 180. p. 1. 
t Luc. cit., p. 15. 
