﻿the Insolation of an Atmosphere. S$;\ 



there. We find 



whence 



i(ft«>-|si±Sri ...... (30) 



This gives the distribution of the emergent- radiation — the 

 law of bolometric darkening. Inserting in (18), the total 

 absorption near the boundary is found to be proportional to 

 (2 — jlog2)S, the emission to |S — i.e. 0*987 instead of 

 unity, an error of only 1*3 per cent. Again, from (19), 

 the net flux at the boundary is given as (| log 2)7rS instead 

 of 7rS — i. e. 1*040 instead of unity, an error of 4*1 per cent. 

 The smallness of these discrepancies shows that (28) and the 

 values (29) are satisfactory approximations. 



To obtain a better approximation, knowing now something 

 of the form of B from (28), we can assume 



B(t) = a-be~ T 



and choose a and b so that the correct net flux is given 

 at the boundary and the condition of radiative equilibrium 

 is satisfied there. It is found that the condition of radiative 

 equilibrium in the far interior is then automatically satisfied, 

 save for terms which tend to zero. We rind 



1(0, 6) = a 



1 + COS0' 



whence from (18) and (19) 



a-/;(2-log2) == JS, 



i a _&(l-log2)= iS. 

 These give 



a= S/log 2 = 1*4427 S, 



b = iS/log2 = 0-7213S; 



whence B = 0*7213S, B»=i-4427S, and 



T„ 4 : I 1 , 4 : T 4 = 1-443 : 1 : 0*721. 



Thus the values of T x and T in terms of T L come out about 

 1 per cent, smaller than on the previous approximation. The 

 relation T oo 4 = 2T 4 still holds. The change is so trifling that 

 we shall not attempt to obtain further approximations, which 

 can be sought by using the integral equation. We shall 



