50 
MR. J. H. .JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
d 2 A 
dt 2 
= V-V' 
Po 
(138). 
Since V' is the gravitational potential of a distribution of density — Ap ( cf. § 6), we 
have 
V 3 V' = 4,it A P , 
(139), 
while if we suppose, for the sake of simplicity, that the motion is adiabatic, so that 
the ratio of pressure to density changes at a constant rate k, we have (cf equation 
(3), p. 5) 
y v = /cvy = - k Po v*a. 
Hence equation (138) becomes 
d~A 
—-7 — 4:irpA — kVA = 0.(140). 
The simplest solution of this is of the form 
where 
A = 
1 
gi (p>t ± qr) 
0 
T 
p 2 + 4:17 p 
K 
( 141 ) , 
(142) , 
and the general solution can be built up by superposition of such solutions. 
Now solution (141) gives A = 0 at infinity, provided q is real, and therefore 
provided qr + ^np is positive, a condition which admits of p being imaginary. 
There is therefore a possible motion, which consists of a concentration of matter 
about some point, the amount of this concentration vanishing at infinity, and the 
amount at any point increasing, in the initial stages, exponentially with the time. 
We conclude, therefore, that a uniform distribution in space will be unstable, 
independently of the mean temperature or density of this distribution. # 
The Evolution of Nebulce. 
§ 47. We can also see that a distribution of matter which is symmetrical about a 
single point will be equally unstable. For, if this distribution of matter were perfectly 
* An interesting field of speculation is opened by regarding the stars themselves as molecules of a 
quasi-gas. If space were Euclidean and unbounded, there would be no objection to this procedure, and 
we should be led to the conclusion that the matter of the universe must become more and more concen¬ 
trated in the course of time. If space is non-Euclidean, this concentration might reach a limit as soon as 
the coarsegrainedness of the structure attained a value so great that the distance between individual 
units became comparable with'the radii of curvature of space. In any case, it may reach a limit as soon 
as an appreciable fraction of the space in question becomes occupied by matter. 
