﻿888 Mr. E. A. Milne on Radiative Equilibrium : 



above should be compared with our formula for a fixed beam 

 'incident at a (from (29")), 



T = T 1 [i(l4-^sec a )]i J .... (33) 



where n = k 1 /k 2 ; and with the rotational mean formula 

 (from (31)), 



10 = ^ [i(l + imr sec X)]*. ... (34) 



Emden's formula differs but little from the latter when \ = 0, 

 as is to be expected. 



Emden does not obtain the integral equation for the 

 temperature distribution. For the sake of completeness 

 it seems worth putting on record the integral equation 

 for the general case involving n and a. It is deduced 

 in the same way as (22) : — 



O'(t) =i»s*-"~ +£ C(T+y)-Q(T-y)^ 



C(t f y) 



i 



2.'/ 



My. . . (35) 



§ 10. Effect of an internal boundary. — We shall next 

 consider the case in which the material is bounded 

 internally by a black surface at t = ti ins'ead of extending 

 to infinity. It has already been mentioned that as the 

 formulae only involve the optical thickness T,we may deduce 

 the results for this case by supposing that immediately 

 beneath t = ti the density suddenly increases indefinitely. 

 The temperature distribution above r x is unaltered. It 

 might at first be supposed that the black surface would 

 assume a temperature equal to T^, but this is not so. For 

 the infinite density gradient we have postulated at the 

 level r x implies an infinite radiation gradient there, and 

 (unless we are prepared to accept the existence of an 

 infinite temperature gradient at the black surface) the 

 surface will take up a temperature intermediate between 

 T^), the temperature of the material in contact with the 

 surface, and T*,. This temperature, say T s , is easily cal-. 

 culated. For since the surface must re-radiate all the 

 radiation falling on it, we shall have o-T s 4 = 7rB s , where 

 B s is given by 



B s = F(ti) + S cos a e - ?lT * sec a ... (36) 



