i9i~i93] Spherical Mass of Gas 195 



The different equilibrium configurations may accordingly be specified by 

 relations of the type 



Different configurations are got by varying a and /"(<?). JFrpm equation 

 (517) it is clear that a single homologous series will be obtained by varying 

 a Avhile keeping f(q) unaltered, while the different homologous series corre- 

 spond to the different functions of q. Thus there are just as many homologous 

 series as there are functions of q } but only those series are stable for which 

 dk/dq is everywhere positive. 



The Condition for Homologous Contraction 



193. The contraction of a configuration under natural conditions will not 

 in general be homologous ; it will be determined by the flow of heat inside 

 the mass. Starting from any assigned real configuration we can calculate 

 the natural changes produced in a mass of gas by the flow of heat and con- 

 sequent radiation in the following way. 



Imagine that each element of the gas is held at rest, and let the natural 

 flow of heat take place for an interval dt, the temperature of the different 

 elements being changed thereby, and therefore the pressures also. Next 

 imagine that each element of heat is constrained to remain attached to 

 its particular element of gas, so that the elements of gas can only change 

 adiabatically, and allow these different elements to arrange themselves in 

 equilibrium under their own gravitation. . The final configuration will be 

 identical with that which would naturally be reached after a time dt. 



During the first process, in which heat flows while the gas is held at rest, 

 the flow of heat per unit area across any sphere may be taken to be /cdT/dr, 

 where K is a coefficient which must always be positive, from the second law 

 of thermodynamics. When the whole transfer of heat is by pure conduction, 

 K will of course be the ordinary coefficient of thermal conduction. 



The aggregate outward flow of heat per unit time across a sphere of 

 radius r, say E r , will be 



j? r --4ir*f~ ........ ...(518), 



or 



so that the rise of temperature at a distance r from the centre will be given 



t>y 



T 9 



' 



This change of temperature will be accompanied by a change in the value 

 of &, and since p is kept constant this will, from equation (514), be given by 



-&' .............................. 'm 



132 



