534 Mr Hargreaves, The Harmonic Expression of the 



the resemblance of which to the integral expression for an 

 associated function, viz., 



I (x + Va,' 2 — 1 cos ty) n cos sty city, 



strikes the eye at once. Physically this corresponds to an 

 absorbing atmosphere in which the fraction of light or heat 

 transmitted is (cosiy -1 , depending on the angle between the 

 zenith and the sun's direction ; one factor cos / being due to the 

 oblique exposure of a surface. 



| 2. First notice that the conditions for arctic and non-arctic 

 regions are embraced in the statement that all values ty between 

 and it, which make sin X sin 8 + cos X. cos 8 cos ty positive, are 

 admitted in the integration. During the period of total day ty 1 

 is continuously ir, and during the period of total night it is 0. 



Now if we expand in zonal harmonics a function which = scP 

 for positive values of x, and vanishes for negative values, the 

 coefficient of P n (x) in the expansion is 



2n + 1 Z* 1 



■"■p, n == 4) "" I M -* ?i y^) MX 



& Jo 



2« + l p(p-2)...(p-n + 2) . I , K \ 



= — ~ — x , i\ , — n \ / ~r\ when n is even } . . .( o). 



2 (p+l)(p+3)...(p + n + l) f 



_ 2n + l (p-l)(p-3)...(p-n + 2) j 



" 2 X (p + 2)(p + 4>)...(p+n+l) " " " ; 



(See Todhunter's Functions of Laplace, pp. 18 and 19.) Introduce 

 this expansion in (4), writing sin X sin 8 + cos X cos 8 cos ty for x, 

 and we get 



2£p )S = I %A P} n P n (sin X sin S + cos X. cos 8 cos ty) cos s^ cfyr, 



Jo 



the integration being now from to it, because we have re- 

 placed the original expression, with its discontinuity and variable 

 limits, by the quasi-continuous expansion. 



With fju = sin X, /a' = sin 8, we have 



P n (sin X sin 8 + cos X cos 8 cos ty) 



\n — s * - 



= 22 ^= (1 - yO 3 (1 - // 2 ) 2 P»« (/*) P.V) cos «*. 



rf s P 



where P,, s (w) = ~~ - , and the factor 2 is to be omitted for 5 = 0. 

 d/ub s 



