Daily Variation of Solar Radiation, etc. 535 



Hence the expansion sought is 



Xv>s = wSi ft n ' '■ = (1 - fl4 (1 - t^4 P n (/*) iV (/*') • - -(6). 



II "T" o 



The formulae quoted for -4p >n , though correct, are rather 

 misleading for the case in which p is an integer ; they suggest 

 that differences in type turn on n being odd or even, whereas in 

 reality they depend on p — n being odd or even. Whether n is 

 odd or even, the group of terms for which p — n is even terminates 

 with n=p, and the formula is 



2n + 1 \P 



2 1 .3 ... (n + p + I) .2 . 4< ... (p - n) 

 When p — n is odd and positive, the form is 

 2n + 1 \p 



2 2.4... (n+p + l).1.3 ... (p-n) 

 When n=p + 1, it is 



2n +1 1 



...(7a). 



.(76). 



2 2^+ 1 (^ + l)* 



When n — p is odd and greater than 1, it is 



2n + l 1 .3 ...(w-jp-2) 



2 2P (2p + 2) (2p + 4) . .. (p + n + 1) 



(7c). 



(_l)*<«^p-i>...(7d). 



Thus up to w=£> + l, all terms odd and even occur and all 

 are positive, beyond this only terms for which n — p is odd occur, 

 alternately positive and negative, the first negative term being 

 that for which n = p + 3. All the terms for which n ~ p is odd are 

 embraced in the single formula 



2n + 1 l.S...(n+p). \p (- 1)* (n+p+^\ », 



~2 2 . 4 . ..(n+j3 + 1) (n- p) (n -p + 2)...(n+p) ' 



which will be found to give correctly the three cases (b), (c) 

 and (d). 



The result (6) was obtained for s = in the previous paper 

 by a much more troublesome method. 



§ 3. Various sequence formulae may be established directly 

 from the definition (4). Obviously 



Xp,« = Xp-m sin ^ sin g + 1 cos X cos 8 {xp-x,s-i + Xp-i.a+i)- • -(8). 



