466 Mr. J. Prescott on Convective Equilibrium 



Now putting n a r for n and n?9 for ± we find 



and 



R_? ■ r >Q. =R jy-^frr+P; . . (10) 

 7 — I i </r x 7 — 1 dr 



r(i)-'-*=ji(f)^-* 



fci+: 



-^ *f(J)^ < u > 



Hence we see that with these values for r lt r , equation (9) 

 is reduced to equation (8) provided 



i3-i 



+ 3o 



n a+8 = n y-i 



i. e., provided 



* + /3 = ^— *+3« (12) 



7 — 1 



This means that, if equation (12) holds, n a r and nfid are 

 the distance from the centre and the temperature respec- 

 tively of an element of gas in convective equilibrium with 



the value ne for -— . Now to ensure that r 1} 6 X shall refer 



to the same element of gas as r, 6 referred to in the old state, 

 we must make the mass within the sphere o\ the same as that 

 which formerly occupied the sphere r. But the expression 

 on the left-hand side of equation (11) is proportional to this 

 mass in the new state. It will be the same mass as that 

 originally in the sphere r if 



n?- 1 =1, 



£=£+3«=0 (13) 



7 — 1 



Equations (12) and (13) give the following values for at 

 and /?, 



