428 REPORT— 1883. 
Opinions such as these have been independently expressed in different parts of the 
world, but no one, the author thinks, has subjected them to the test of calculation. 
It is rather curious that, as far as he is aware, the problem of the internal 
equilibrium of a gaseous gravitating mass has not as yet been discussed. Such a 
mass will arrange itself in concentric layers round its centre of inertia, and the 
question arises, What is the distribution of density and pressure within? It is 
exceedingly probable that the problem has been attempted, for it is perfectly easy 
to write down the differential equation which contains the result. But then the 
differential equation has to be solved. If we suppose the temperature constant 
throughout the gas, we cannot express the result in closed form; if the temperature 
is regulated by the adiabatic law, we must take account of the ratio between the 
two specific heats. For two different values the equation can be solved, and as the 
ratio for all known gases happens to lie between these values, we may at any rate 
get some information from a consideration of these special cases. If the ratio 
between the two specific heats is 1:2, the following system of equations gives the 
result :— 
27 ec 
P= "A5 Qrg)i (c2+r2)3 
9 sy Cc? 
Pauad (23rg)* (ce? +77)3 
6 J/3 Sc ? 8 
lV fia POR Ga ae ae 
5 Agi (27) (c? +77)2 
Here p represents the pressure; p, the density; 7, the distance from the centre of 
mass; g, the constant of gravitational attraction ; M, the total mass within sphere 
of radius and round the centre of mass; ¢ is a constant of integration to be deter- 
mined by the conditions of the problem, and 4 depends on the nature of the gas 
(p=Ap). The total mass of the gas is BA8 lB t as we see by putting r 
Aig (27)*, 
infinitely large in the last equation. 
The gas extends to an infinite distance, as only there the density and pressure 
vanish; the distribution of temperature is, of course, also determined by the 
equations. 
The second case, for which the differential equation is easily solved, is that in 
which the gas has its specific heat for constant pressure exactly twice that of con- 
stant volume. With the same notation as above, only putting a=A ./27ry, 
cue 
/p= -smar 
, 
Ges 
p=—— sinar 
, 
2e 
M=— (sinar—a7cosar). 
ga © ) 
Here the mass of gas is limited, for the equations kave only sense as long as a7 
remains between o and 27. When, therefore, 7» has become equal to ~ , the 
“7 
4c 
gA- 
It would be interesting to inquire for what value of & (the ratio of the two specific 
heats) the mass begins to arrange itself into a finite sphere. The author believes 
that this occurs when this ratio is equal to *, but cannot offer any absolute proof. 
The highest value which / can have is 12, and this value holds approximately for 
mercury vapour; it is very probable that the value of x in the interior of the sun, 
where molecules will no doubt be broken up, as far as they can be broken up by 
heat, will not be far from the same number. To follow out the calculation in this 
pressure and density are nothing, and the total mass is equal to 
