Daily Variation of Solar Radiation, etc. 533 



the distance of the sun and the angle of exposure to the sun's 



rays. The element of heat-supply may be written — — cos /, 



in which dt is time-element, r the distance of the sun from the 

 earth, and / the sun's zenith distance. In latitude X, when the 

 sun's declination is 8, cos / = sin X sin 8 + cos X cos 8 cos ty ; ty the 

 hour-angle varies in the course of the day from — ty x to + ty 1 , 

 where ty 1 the hour-angle at sunset is given by 



sin X sin 8 + cos X cos 8 cos ty± = 0. 



The factor cos / is positive during the day, and at night is replaced 

 by zero. 



In lieu of this discontinuous expression, we propose to write 

 the quasi-continuous expansion in cosines of multiples of ty, in 

 which ty varies from — ir to + ir, embracing the whole twenty-four 

 hours. Fourier's method gives for this 



-^-Xxscoss^ (1), 



in which 



^ s = I (sin X sin 8 + cos X cos 8 cos ty) cos sty dty . . .(2). 

 Jo 



The factor 2 is to be omitted for the opening term (,§ = 0), 

 on which depends the day's integral heat-supply, and so the 

 element of the annual variation discussed in the previous paper. 

 The other terms express the daily variation divided into harmonic 

 components, whose periods are the day and its exact sub- 

 multiples. 



Integrating by parts, for the general case ^ 



_ cos X cos 8 /sin (s — l)ty 1 sin (s + l)-v^A 



Xs ~ ~2s [ s-1 ~s~+l ) 



and for the case 8 = 1, 



cos X cos 8 . . 



%i = « (^i - i sin 2^) 



.(3). 



As ty 1 is known in terms of X and 8, these formulae give 

 numerical values of the coefficients for an assigned latitude and 

 date of the year. But a general expression for ^ s may be obtained 

 in a series of associated Laplace's functions with sin A. and sin 8 as 

 arguments, applicable to the whole globe and taking proper account 

 of the discontinuity that occurs in arctic regions. For greater 

 generality this is found for 



Xp,s — I ( sm ^ sm & + cos X cos 8 cos ty) p cos sty dty. . .(4), 

 J o 



