706 Lord Kelvin : The Problem of 
behaviour of the Boylean gas from that of gases for which 
k< 3, and the resemblances of the Boylean gas and of gases 
for which /e>3 (of which it may be regarded as the limiting- 
case, k=co), 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 con- 
ditions of a mass of any gas for which /c>3, and one 
equilibrium condition of a mass of any gas for which /c<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 of gravity. 
§ 40. Let K r 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 = i7rK v \drr 2 pt = K v \ dmt . . (42). 
Jo Jo 
By using equation (6), and then integrating by parts, we 
obtain 
I= 47rR 
f<<"'^[(M;-li>43 w. 
and since p = at the outer boundary of the sphere and r = 
at the centre, we have 
Substituting now the expression given for — -j- in the 
equation of hydrostatic equilibrium (34), we obtain finally 
i=, .^jT* ,, * w (45) - 
§ 41. The work which is done by the gravitational attraction 
of the matter within any layer of gas ±irr 2 pdr in bringing 
that layer from an infinite distance to its final position in the 
sphere is given by 
dw — \nrv 2 pdr.gr (46); 
and the work done by gravity in collecting the whole sphere 
