o8 Trans. Acad, Sci. of St. Louis. 



the integral of this from r^ to r is 



r 



^, =^-itf„ — ^=-fci\/„-^. (16> 



This work is quite independent of any change of tempera- 

 ture. 



On the other hand, the work of compression is largely 

 dependent on the temperature. Evidently the work of com- 

 pression is 



V r*V 



E= \p{ -dv)=0 1=^, (16) 



■ ip(-dv)=C i^ 



J ^0 J^o 



in which, in general, T will vary. Now in order to deter- 

 mine whether the temperature must fall to make contraction 

 possible, or rise as the consequence of an excess of energy in 

 the force of attraction, let us first suppose the contraction to 

 be isothermal, i. e., let the heat produced by the compress- 

 ing force be in some way radiated out of the system as fast 

 as generated, so that the temperature (and hence the energy) 

 of the gas is all the while constant, with T= T^. Equation 

 (16) thus, by (14), becomes 





*v 

 E=CT, I =^-=c'r^iogl«=30r„log^«. (17) 



In order to compare (12) with ( 14), substitute the value of 

 M^ in (11), and we have 





E,= ZaTAog'^- 



(18) 



