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



We shall now prove that I(f 2 ) =2B(* 2 ). From (9), re- 

 membering that B is constant in (7 2 , 0), we have 



I(* 2 ) = J* ( h B(t } e-hlt ^(t^e-it^-t-i), 



1(0) = j tl ¥>{t)e-m + B(t 2 ){l-e-h) ^K^e-h, 



whence 



l(0)-e-t-2l{t 2 ) = B (*,)(! -*-*). 



Further, from (10), 



r(^) = B(; 2 )(i-.-* 2 ) (i3) 



Inserting in the equation for t 2 , namely (8), we find 



ife)(i-«-*) = 2B(y(i~*-V), 



which is the equality required. Making use of this, we have 

 from (11) and (12) 



Eft) = IU) = -B(0). .... (14) 



Now I' is essentially positive. Hence there is an excess 

 of absorption over emission at the base and a numerically 

 equal excess of emission over absorption at the top. This is 

 the result stated. The excess can only be zero if I'{t 2 ) 

 is zero, i. e. if t 2 is zero. 



It should be noticed that the departure from radiative 

 equilibrium at the base and at the top is very appreciable. 

 The ratio of the excess, 2tt kpdv'E(t 2 ) i to the emission, 

 4LirkpdvB(t2), has the value 



i(i-*-'0; (15) 



if £ 2 =1'0 this is 0'32, and if £ 2 = 0\56 it is 0'22 ; and it can 

 be shown from Gold's data that these limits for t 2 correspond 

 to widely separated values of t u the total absorbing power of 

 the atmosphere. Again, E(£) is a continuous function of t; 

 and hence, since it is positive when t = t 2 , it will be positive 

 in the upper parts of the convective atmosphere, violating 

 the condition for convection. As we approach the earth 

 it decreases, soon becoming negative, showing that in the 

 lower portions the condition is satisfied. 



We have assumed the atmosphere " grey " as regards the 

 low-temperature radiation, and we have ignored the direct 

 absorption of solar radiation. But a variation of the co- 

 efficient of absorption with wave-length does not affect the 

 gist of the argument ; a strictly isothermal upper atmo- 

 sphere would still be an impossibility unless its optical 



