AND THE OTHER PLANETS OF THE SOLAR SYSTEM. 669 



35. We know too little of Mercury to form any judgment as to the possible effects of 

 any known causes in modifying the intense heat of radiation to which he must be subjected in 

 consequence of his proximity to the Sun. 



36. In considering the case of the Moon it will be sufficient for my purpose to suppose 

 her axis of rotation to be perpendicular to the plane of her orbit, and that plane to pass 

 through the Sun. I shall moreover restrict myself to the effect of the solar heat on a place 

 situated on the Moon's equator. The problem may be solved by means of the general 

 formula; given by Poisson, but so far as regards the numeral quantity Q, it may be more 

 simply solved as a particular case. 



Let 9 = the zenith distance of the Sun at any proposed place on the Moon's equator, 



S 

 at any time t ; and let — denote the intensity of solar radiation at the distance a. It will be 



sufficient to consider the Sun's distance from the Moon as constant, and equal to the Earth's 

 mean distance from the Sun. Then shall we have 



£.£ + __ CO8 0, 



= f + irh cos 9 (Art. 17) ; 

 £ being the temperature of an imaginary medium surrounding the Moon (Art. 17), and f 

 now denoting the temperature derived from stellar radiation alone, the Moon being assumed 

 to be without sensible atmosphere. Now the last term will have the value wh cos 9 only 



so long as the Sun is above the lunar horizon of the proposed place, i. e. from 9 = 



2 



7T 7T 



to 9 = -, and will be equal to zero from 9 = — to 9 = n. It will therefore be a 



periodical discontinuous function. But any function, f{^/), of this kind, can be expressed 

 by the well-known formula 



/(f) = ^jT/Wty' + I 2 {£ cos i ty - f')f^df'\ ; 



and hence putting /(^) = cos 9, and taking the integrals only between - - and - since 



2 2 



the other portions vanish, we have 



ttA cos 9 = - jl cos ffdff + A2 { J\ cos i (9 - 9') cos ffdff \ , 



the last term including all values of i from unity to infinity. This expression gives the 

 value of h cos 9 subject to the condition that it shall equal zero while the Sun is below the 

 horizon of the proposed place. Performing the operations indicated, and taking all values 

 of t from unity to infinity inclusive, we have 



n C os0 , d0'=2, 



IT 5 * 



f 1 - • /a c/\ of j/v /sin (t + 1)3 sin (i + l)*\ 



/ . cos i (9 - 9) cos ffdff = I — H — + — 7 cos %9 ; 



J ~2 \ * + 1 * - l / 



