278 
Proceedings of the Royal Society of Edinburgh. [Sess. 
I/W would therefore go on diminishing instead of increasing, as it would 
require to do if the gas is to be restored to a condition of equilibrium. 
§ 45. “ From this we see that if a globe of gas Q is given in a state 
of equilibrium, with the requisite heat given to it no matter how, and left 
to itself in waveless quiescent ether, it would, through gradual loss of heat, 
immediately cease to be in equilibrium, and would begin to fall inwards 
towards its centre, until in the central regions it becomes so dense that it 
ceases to obey Boyle’s Law ; that is to say, ceases to be a gas. Then, not- 
withstanding the above theorem, it can come to approximate convective 
equilibrium as a cooling liquid globe surrounded by an atmosphere of its 
own vapour.” * 
§ 46. But if, after being given in convective equilibrium as in § 45, 
heat be properly and sufficiently supplied to the globe of Q gas at its 
centre, the whole gaseous mass can be kept in the condition of convective 
equilibrium. 
§ 47. The theorem of §§ 42, 43 is given by Professor Perry on page 252 
of Nature for July 13, 1899 ; and in the short article “ On Homer Lane’s 
Problem of a Spherical Gaseous Nebula,” published in Nature, February 14, 
1907, I have referred to it as Perry’s theorem. Since this was written, 
however, I have found the same theorem given by A. Ritter on pp. 160-162 
of Wiedemanns Annalen, Bd. viii., 1879, with the same conclusion from it 
as that stated in § 44 above, namely, that when k< 1J the equilibrium of 
the gaseous spherical mass is unstable. 
§ 48. In the theorem of Ritter and of Perry, given in section 42, con- 
vective equilibrium is not assumed. For the purposes of our problem in- 
dicated in § 21, it is desirable to obtain expressions for the energy and the 
gravitational work of a mass M in equilibrium with a stated density at its 
centre, in terms of the notation of §§ 23 ... 25 above. Thus, taking as our 
solution with central temperature C (equation 26), 
t = C©(z) (50), 
where 
Z = X C~^-V ; r = <rC~ i(K - 1 ) /z; 
and where cr is given in terms of the Adiabatic Constant, A, by (22); we 
have from equations (25) and (50) 
to = ( k +1)SjtC-^- 3) @ ,( z ) .... (51), 
and by differentiating this we obtain — 
dm_( K + l)ScrC-ti- 3) 
E e 2 
®"(z)dz 
(52). 
* Quoted from “ On Homer Lane’s Problem of a Spherical Gaseous Nebula,” Nature , 
Feb. 14, 1907. 
