﻿the Insolation of an Atmosphere, 887 



Tj by the relation 



T = Tx[i(l + *,/«]* (32) 



where k A and /r 2 are the coefficients of absorption for the solar 

 and terrestrial radiations. 



Both these results are in apparent contradiction with those 

 obtained in this paper. The source of the discrepancies is in 

 each case Emden's assumption that the incident radiation 

 may be taken to be diffuse. The way this occurs is as 

 follows : — The mean coefficient of absorption for diffuse 

 radiation incident on a thin layer of material is approxi- 

 mately twice the coefficient of absorption for a parallel 

 beam incident normally, i. e. twice the coefficient of ab- 

 sorption as ordinarily defined. This fact allows us to 

 approximate to the equations of transfer (equations (1) 

 and (2) above) by replacing them by the " equations of 

 linear flow " ; in equations (5) and (6) we have explicitly 

 adopted a new optical thickness t equal to twice the optical 

 thickness t obtained directly from the ordinary coefficient 

 of absorption ; in equations (24) and (25) we have retained 

 the optical thickness t and simply replaced the factors cos 6 

 and cos-^r by the value ^ ; the result is the same as if all 

 the diffuse radiation were supposed to be confined to beams 

 at an angle of incidence of 60° with the planes of stratifi- 

 cation. Emden approximates in the same way as we have 

 done, but since he takes the solar radiation to be diffuse 

 he is adopting for this also a coefficient of absorption twice 

 the value for a permanent beam. His results may therefore 

 be expected to agree with ours if in ours we put cos a. = -J, 

 a — (i0°; and this in fact they do. But they lose part of 

 their significance. His result for case (1) is of course 

 true for diffuse radiation ; indeed it is obvious thermo- 

 (lvnamicallv, without proof, that material exposed to iso- 

 tropic incident radiation will, if in radiative equilibrium, 

 take up a temperature equal to that of the radiation : the 

 case is practically that of a black body enclosure. But 

 our results show that if the incident radiation occurs as 

 a parallel beam — as, in fact, solar radiation does — then 

 the isothermal state is merely the particular distribution 

 of temperature that happens to correspond to an angle 

 of incidence of 60°. Further, Emden's result does not 

 suggest another of our results — that when n^fil there also 

 exists an isothermal state of equilibrium : namely, for 

 cosa = ^/ for a fixed parallel beam, and for cos\=>?/\/2 

 when rotation is taken into account. Emden's formula (32) 



