jEj.vpressiuo Sohitions of Laplace^ s Equation. 207 



Hence 



Now substitute iu this for J J from (45) and use (43), 



n— 2 



»? 



P- + v^k^/ip;:-^ + .77;^ ^:r - (i-^^)p:u (49) 



-^(,,_l)- " ^ ,,_i '-^ n{n-l) 

 But from formula (46) 



/1 — 2 p'"+^ p"^ n — m — l -^nz 



^ ^ '^ 91 — 1 



Substituting into (49) for P;"+/, P^^,^, it becomes, after 

 reduction, 



(n + ?n) (?i — m)Fn + ^/i (^^ — l)Fn-2 — fjin{2n — 1) P^_i = 0, 

 or in terms of P''s, and writing 7^+ 1 for 71, 



(^n-mi-l)V:\, + in-hm)F:Li=fM{27i+l)T':, . (50) 



another well-known recurrence formula which includes, when 

 m = 0, the recurrence formula for Legendre functions. 



16. The formulae invohdng the derived function of the 

 ordinary Legendre function P„(/x) can be obtained in a similar 

 manner. 



dTn d y / / d \ , 



lOW 



D(,/(/.)D/=D^/(M)rD" 



