56 Trans. Acad. Sci. of St. Louis. 



heuce the integral of (6) is 



2 C'T^^ 



(9) 



the constant of integration being obviously zero. 

 Substituting this value of ^ in (4) we have 



rfM=^^", (10) 



which means that all spherical shells, in a given mass of 

 perfect gas in equilibrium and at uniform temperature, have 

 the same mass if they have the same thickness, whatever be 

 the radius, their centers being always at the center of mass. 



Integrating ( 10) from r = to r = r^, we have 



M^^'-^'^'Jf:^. (11) 



From (9) we see that at the center of mass where r = 0, 

 we have p = Oj ; and that^; = only when r = CO . Hence, 

 whatever be the total mass of the gas, it extends to infinity 

 under the conditions assumed. The general value of 31 is 



c 



M=^-^. (12) 



which is infinite when r is infinite; from which we infer, that 

 if there is at any isolated point any manifestation of a perfect 

 gas in the way of pressure, density or temperature, the total 

 unlimited mass of such gas must be infinite. On the other 

 hand, if the total mass is finite, it can only exist with its 

 temperature zero. 



The uniform temperature of our mass has been T^. Equa- 

 tion (12) shows that had the temperature been higher, say 

 Tj the mass inclosed in the same sphere of radius r would 

 have been proportionally greater ; that is 



M T 



— = ±0. (13) 



