spherical Wave of Light. 75 



On differentiating, for example, — ^ let us write the result 



-——, ~, and let us write the subsequent numerators re- 



cr ^"+1' 



suiting from successive differentiation Uo, 21"' etc. 



Also let us write 



u = w + vy- -\-ty^ + sy^' + ?y^ + etc. 



and differentiate/ times, using Leibnitz' Theorem. 



ziP={- iY{i^' - 2pv^~\x + c) -p{p - i)z:^'-' 



+ 4/(/ - i)/^-> + cf + AP{P- i)(/ - 2y-\x + c) +p-(p - 3)F-' 



- Sp--{p - 2)sP-%x + cf - 12P- -(p - s)s^-\x + cy 



- 6p-{p - 4)s^-\x + c) -p- - -{p - 5)s^-' + etc. } 



in which j/ is put zero. 



We are now in a position to do the formal integration, 



as a first step we have the formulae 



(n - 2) — (7i+i)Uo (u - 2){n — 1)1^2 t/i 



2^V-^ 2rr"-^ 



(n-^y-mio ' _ {n - 3)(?i - 2)212 uj 



(;/-3)---(;g + 2)?/, _ (;/-3)---;/?^2 (;^ - 3)(^^ - 2)2^4 u^ 



48^'*+^ i6r"+^ 16/-"-^ 48r"-5 



In calculating this expression the same kind of series 

 continually presents itself. We may write this series in the 

 general form 



n{7i+ iy--{n + f- i) -p{n + s){;i + s + i)-—{7i + s + t- 1) 



^PiP '(^f^ + 2s)(n + 2^ + i)- - -(;/ + 2S + / - i) - etc. 



Writing 



n(7i -f i) — {71 + f- i) = ?/,(, and E for i + A 



