146 Trans. Acad. Sci. of St. Louis. 



and heat is being taken from the gas. For isothermal com- 

 pression, the specific heat is of course infinite. In the cases 

 lying in the angle t, in, p, compression is still going on, but 

 the temperature is falling, while heat is being removed from 

 the gas. In the angle ^, m, v' , both temperature and pres- 

 sure are falling, and the specific heat is still negative. It is 

 evident that the gas at any point in a gravitating nebula, 

 must be going through an operation which would be repre- 

 sented by a line in the figure lying within the angle «, 7n,p. 

 We shall make an attempt to locate this path. 



At a distance R from the center or nucleus of a cosmical 



'• gaskugel," where the density of the gas isS=-, the 



mass internal to R being M, the law of gravitation gives the 

 equation 



dP _ ^ 



where k is the gravitation constant. 

 By eq. (3) this becomes 



1 



dP _ M/PY 



dR ^r\Aj 



Hence 



k dR 



.A\^ R^dP 

 M 



-(I) 



By differentiation, 



1 

 dM_ ^n 



dR^ — J 

 By Geometry 



1 — « 



dP. -1 



B^ d^ , dP( iPnR ^^^ " dR 



'^^ dm "^ dR\ ' ~i 



1 



dM , Pn 



-^ = 47ri?2S = 47ri?2 — ^ 



A" 



