1907-8.] The Problem of a Spherical Gaseous Nebula. 
289 
APPENDIX. 
By George Green. 
§ 1. In order to determine the conditions of temperature, pressure, and 
density, at any distance from the centre of a spherical mass of gas in 
convective equilibrium, held together by the mutual gravitation of its 
parts, it is necessary to find a solution of the equation, 
dH__ 
dx 2 — 
(!)■ 
In this equation, 
x is inversely proportional to r, the distance of any point from the 
centre of the sphere : 
r — 
x _ 
t is the temperature of the gas at any point of a spherical surface of 
radius r : 
t K is proportional to the density where t is the temperature : 
y- is proportional to the mass of gas within the surface of radius r : 
OiOfy 
~dt e 2 ml 
_dx (k + 1)Sct eJ 
and k is equal to l/tfc— 1), where k is the ratio of specific heats of the 
gas considered (see §§ 22 ... 24 of the above paper). 
§ 2. Solutions of this equation can be found which correspond to a mass 
of gas around a solid or liquid nucleus. These may become of interest 
later. Solutions can also be found which correspond to an infinite 
sphere of gas, with an infinite density at the centre. But the solutions 
which it is now desirable to obtain are those which can be applied to the 
case of a spherical mass of gas which has a finite density at the centre. 
This is expressed mathematically by saying that at x — oc, we have 
* = C, 
dt 
dx 
0 . 
Lord Kelvin has shown, in his paper “ On the equilibrium of a gas 
under its own gravitation only,” Phil. Mag., March 1887, and in § 25 of the 
YOL. xxvm. 19 
