﻿the Insolation of an Atmosphere. 881 



From (16) and (17), with the appropriate boundary con- 

 ditions, 



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I(r) = e T8ec0 \ B(*>"' 8CC * sec dt, . . (20) 



I'(t) =«" TBeof *l B(t)e tsec * sec <fdt. . . (21) 

 Jo 



Substitute these in (18) ; write t = r + y cos 6 in the first 

 integral, t = r— y sec ijr in the second, and replace cos# 

 and cos x/r by /a. We find 



J ^ B(t +#/*)*-% 



Jo Jo 



+ fV ( Tl K(T-yix)e-ydy + ^e- T = 2B(r). 

 Jo Jo 



We can now reverse the order of integration in the repeated 

 integrals *. Setting 



=f 



C(t) = B(r)dr, C'(t) = B(T), 



we find finally for the integral equation for the temperature 

 distribution. 



Jo fy 



e v d y 



fC(r + 

 Jr 2y 



^e-vdy (22) 



If we invert the orders of integration before making the 

 substitutions for t, we obtain another form, 



*(-r)=irB(t)Ei(\t-T\)+i&e-T, . . (23) 



Jo 



which is the standard form for integral equations t- 



Solutions of these may be sought directly. For an 



* For details, cf. Monthly Notices, lxxxi. p. 365 (1921). 



t In equation (23) Ei denotes the exponential-integral function. 

 The integral equation in the form (23) is substantially equivalent 

 to the integral equation obtained bv L. V. King in the analogous 

 problem for scattering (Phil. Trans." 212 A. p. 375, 1912); it bears 

 the same relation to King's equation that the author's integral 

 equation for the atmosphere of a star in radiative equilibrium 

 (M. N. lxxxi. p. 373, 1921) bears to Schwarzschild's integral equation 

 for scattering in a stellar atmosphere {Berlin Sitz. 1914, p. 1183). 

 But the form (22) is more convenient when solutions are being sought 

 by successive approximation, and for other purposes. 



