196 The Evolution of Gaseous Masses [OH. vm 



In the second part of the motion, the various elements all change adia- 

 batically, so that k remains the same for each. Thus the whole rate of change 

 of k for the element initially at a distance r from the centre will be given by 

 equations (520) and (519), or by the single equation 





This equation accordingly determines the rate of change of k, in the actual 

 motion, for any element of the mass. Knowing the rate of change of k we 

 know the values of k at the end of the interval dt, and these suffice to deter- 

 mine the whole configuration. 



194. The final configuration will be homologous with the original one if 

 the initial and final values of k for every element are connected by a relation 

 such as (517). Thus the condition that the contraction shall be homologous 

 may be put in the differential form 



where f is a constant throughout the mass. Comparing with equation (521) 

 we obtain the condition for homologous contraction in the form 



The total heat-content inside a sphere of radius r may be taken to be 



H r = [* 



JO 



Using this relation and (518), equation (523) becomes 



dEr__ dHr 



dr ~ * dr ' 



At the centre E r and H r both vanish ; at the surface they become equal 

 to E and H. Hence, eliminating f, the condition for homologous contraction 

 may be put in the form 



or, replacing E r by its value, 



?-* ........................... (525). 



The Permanent Homologous Series 



195. If equation (525) holds throughout the life of a mass of gas, its 

 whole motion will be along a single homologous series. The equation must 

 of course be true for all values of a and q. Now during homologous con- 

 traction r*dT/dr is, from equation (512), a function of q only, as of course is 

 also the fraction H r /H. The total emission E is a function of a only. 



