276 Proceedings of the Royal Society of Edinburgh. [Sess. 
become of interest when we come to the question of the possibility of 
equilibrium of a mass of gas which is gradually losing energy by radiation 
into space. The result found above that there are two equilibrium conditions 
of a mass of any gas for which k > 3, and one equilibrium condition of a 
mass of any gas for which k < 3, within a given sphere, makes it desirable to 
investigate the nature of the equilibrium in each case, and leads us to the 
consideration of the energy required to maintain a mass of gas in equilibrium, 
within a sphere of radius R, in balancing the condensing influence o 
gravity. 
§ 40. Let K v denote the thermal capacity at constant volume of the 
particular gas considered. The energy within unit volume of the gas at 
temperature t is K v pt ; and the total energy I, within a sphere of radius R 
is given by 
I = 47rK v ^ dr r 2 pt = K v J dmt .... (42). 
o o 
By using equation (6), and then integrating by parts, we obtain 
and since p — 0 at the outer boundary of the sphere and r = 0 at the centre, 
we have 
dr 
(44). 
Substituting now the expression given for 
static equilibrium (34), we obtain finally 
dp 
dr 
in the equation of hydro- 
47rK 
J = dr r*gp (45). 
0 
§ 41. The work which is done by the gravitational attraction of the 
matter within any layer of gas 4t7rr 2 pdr in bringing that layer from an 
infinite distance to its final position in the sphere is given by 
die = 47 T7' 2 p dr. gr . . . . . . (46) ; 
and the work done by gravity in collecting the whole sphere of radius R 
is therefore 
W = 47 r ^ drr d gp = e ~ j dm— . . . . (47). 
0 u 
§ 42. From equations (45) and (47) we obtain, as the ratio of the 
