Gaseous Stars 
68 
[oh. m 
where I stands for Sm (a? + y 2 + z 2 ), in which form it is not restricted to states 
of steady motion. 
If, as before, /3 denotes the ratio of the total internal energy of the 
molecules to their energy of translation, the total heat-energy of the gas is 
(1 + /3) T, and the total energy E is given by 
so that E increases with T if /3 is greater than unity, but decreases with T 
increasing, if /3 is less than unity. This brings us directly to the results 
already given in the last section. 
62. We can write the kinetic energy T in the form |2mfl 2 , where v is the 
velocity of translation of a molecule of mass m. The potential energy W may 
similarly be written in the form — ^2mF, where F is the gravitational 
potential at the point occupied by the mass m. Thus Poincare’s theorem 
takes the form that 
so that, in the steady state, the average value of v 2 , averaged over all the 
separate masses, is equal to the average value of \ F 
If the system is of total mass M and has a mean radius r, the average 
of v 2 is of this order of magnitude. This provides a convenient rough measure 
of the average velocity of agitation of a system of gravitating masses in a 
steady state: it is equally applicable to systems of stars, star-clusters, nebulae, 
and masses of gravitating gas. 
If the particles which constitute the system are taken to be the molecules 
of a gas, or other independently moving units such as atoms, free electrons, 
etc., v 2 is equal to SR/mp times the temperature of the gas, where is its 
mean molecular weight. Thus the mean temperature of the gas is of the 
order of magnitude of 
so that the mean internal temperatures of different stars are approximately 
which iH=2xl0 33 , r = 6 , 95xl0 10 ) is supposed to be formed of hydrogen 
molecules for which fi — 2, its mean temperature must be of the order of 
15,000,000 degrees; if it is formed of molecules of air, the mean temperature 
will be about fourteen times this, or 210 , 000,000 degrees. 
E = (1+/3)T+W 
In the steady state in which 2T + W = 0 it follows that 
(61-8). 
T(/3-l) = E 
(6P9), 
2m (V — |F) = 0 
(621), 
value of ^F is of the order of magnitude of 7 Mjr, so that the average value 
(62-2), 
proportional to the values of ¡¿M/v for these stars. Lane’s law is included as 
a special case. 
As regards absolute values, we find from this formula that if the sun (for