2 77 
1907-8.] The Problem of a Spherical Gaseous Nebula. 
intrinsic energy within the sphere of gas to the work done by gravity in 
collecting the whole mass from an infinite distance, 
W 3S 
• (48). 
If K p be the specific heat of the gas at constant pressure, we have 
S = Kp — K v , and equation (48) may now be written in the form 
I K v I k 
W~3(K P -K V )~ 3(k — 1) 3 
(49). 
§ 43. According to this theorem, it is convenient to divide gases into 
two species : species P, gases for which the ratio (k) of thermal capacity 
pressure constant to thermal capacity volume constant is greater than 1J ; 
species Q, gases for which k is less than 1-J-. And the theorem expressed 
mathematically in equations (48) and (49) may be stated thus : — “ A 
spherical globe of gas, given in equilibrium with any arbitrary distribution 
of temperature having isothermal surfaces spherical, has less heat if the gas 
is of species P, and more heat if of species Q, than the thermal equivalent 
of the work which would be done by the mutual gravitational attraction 
between all its parts, in ideal shrinkage from an infinitely rare distribution 
of the whole mass to the given condition of density.” * 
§ 44. It is easy to show from the theorem of §§ 42, 43 that the 
equilibrium of a globe of Q gas is essentially unstable. Let us first suppose 
for a moment that by a slight disturbance of the equilibrium condition the 
ratio I/W for the globe of Q gas becomes greater than that required for 
equilibrium by equation (49). Unless the excess of internal energy were 
quickly radiated away, the repulsive force which the globe of gas possesses 
by virtue of its internal energy would more than balance the condensing 
influence of gravity, and the globe would tend to expand. Since the internal 
energy lost in expansion is exactly equivalent to the work done against 
gravity, we see that the ratio I/W would continue to increase and the globe 
would become farther from an equilibrium condition than before. The 
expansion of the globe would therefore go on at an ever increasing speed 
till the density of the gas becomes infinitely small throughout. 
If, on the other hand, through a slight disturbance of the equilibrium 
condition, the ratio I/W becomes less than that required for equilibrium, 
the globe of gas would in this case tend to contract. The increase in the 
internal energy due to any slight condensation would be exactly equal to 
the thermal equivalent of the work done by gravitation ; and the ratio 
* Quoted from “ On Homer Lane’s Problem of a Spherical Gaseous Nebula,” Nature , 
Peb. 14, 1907. 
