Daily Variation of Solar Radiation., etc. 537 



or 



(11), 



a sequence equation in which only the letter p changes. 



The elimination of x P ,s from (8) and (9) gives an equation in 

 which only p — 1 occurs, and if we raise by 1 the value of p this is 



£ cosX cosS {(s+p + 1) x P ,s+i + -p-l) Xp,s-i] + sXp.s sin X sinS = 



(12), 



the sequence equation in which only the letter 5 changes. 



Further differentiating x P ,s with regard to X, we easily obtain 



ay 

 cosX -*£' s +px Pt8 sm'K=pxp-i,sSmB (13), 



and differentiating again 

 cos X ^- s + (p - 1) sin X 4gi +p Xp , s cos\=p sin 8 ^^ . 



Multiply this by cos X, and substitute for p sin X cos X -^- s 



and for p cos X sin S Ji^~ from (13) (in the latter case with p 



reduced to p — 1 ), then 



-> d 2 x P ,s ■ . . dx P ,s . ,. 



cos- X Cr - sin X cos X ^r + ^% ;)i s cos- X 

 ftX ciX 



=■ p 1 sin 2 X - p (2p - 1) Xp-i,« sm ^ sm <> + P (i J - 1 ) Xp-2,s sin 2 S. 



Simplify the right-hand member by the sequence equation 

 (11) and divide by cos 3 X, 



d\ 2 



The left-hand member of this equation has the form of 

 Laplace's equation for ® PtS . In this case the function is linked 

 to the alternate one by the right-hand member, which is zero for 

 the Laplacian. 



§ 4. The result (6) for the case in which sin 8 or // = 0, gives the 



expansion of (1 - /r)^ in associated functions, an extension of the 

 well-known expansion of (1 — /a 2 )* in zonal harmonics. With 

 /*' = 0, 



T2 7T. IjOCOS^X 



X,,,, -eo#XJ ^ co^ f . cos s f d* -pffgg^jgf+i) 



-tan\^-' + |j,(p + l)-^ ; | Xp ,,=y(p-l) X!M!i , (14). 



