1907-8.] The Problem of a Spherical Gaseous Nebula. 281 
concentric with the boundary, is constant. We have accordingly for the 
mass m 
! Z ' C ' 
(63), 
where C varies slowly as time goes on. If we suppose C x to be the initial 
central temperature of the nebula, and C 2 its central temperature after a 
quantity of heat H has been lost by radiation, by applying (62) in the 
equations given above we easily find (with suffixes 1 and 2, referring to 
the initial and final conditions respectively) the following results : — 
a 
t ^~C 1 tl 
Cj 
2 
c,'* 
Pi 
R, 
2 - c, Rj 
±2 c, il 
r 2 = C] r l 
w 2 = ^w I 
(64); 
in which t 2 , t v p 2 , p v r 2 , r v all refer to points on the spherical surface enclosing 
a stated mass m. The total heat lost by radiation may now be written — 
H = (W 2 -W 1 )-(I 2 -I 1 ) = 
^2 ^1 
C, 
(Wx-I,) • 
(65); 
and for an infinitesimal change in the condition of the whole mass at any 
time this becomes 
SH = -q (W - I) . .... (66). 
§ 53. These are interesting results. Remembering that I x = k/ 3.W l5 we 
see by (65) and (66) that the central temperature of a globe of gas P in 
equilibrium increases through gradual loss of heat by radiation into space. 
We then see also by (64) that the internal energy of a globe of gas P, 
continuing in a condition of approximate equilibrium while heat is being 
radiated away across its boundary, would go on increasing, and the work 
done by the mutual gravitation of its parts would go on increasing, till the 
gas in the central regions becomes too dense to obey Boyle’s Law. At 
the same time, the radius of the globe would diminish. In other words, 
the repulsive power which the globe of gas P possesses by virtue of its 
internal energy, while in approximate equilibrium, is, owing to gradual 
loss of energy by radiation, at each instant just insufficient to exactly 
balance the attractive force due to the mutual gravitation of its parts. 
The globe is therefore compelled to contract ; and, as the heat due to the 
contraction is not radiated away so quickly as it is produced, the shrinkage 
•of the globe is accompanied by augmentation of its internal energy. 
In figures 1 and 2 curves are shown illustrating five successive stages, 
