6 Prof. J. W. Nicholson on Zonal 



Thus 



f P 2 n + i(/*)Q2m(^)^= - 1 / (2m - 2n-l)(2m + 2n + 2) 

 Jo ... (14) 



for all integer values of m and n. In the same way, 



PP2»M Q*»+iM ^= - 1 / (2m- 2n + 1) (2m + 2n + 2), 



. '. (15) 







and the fourth case can be worked out at once 



Expansions of Harmonic Functions. 

 If, returning to (4), we have 



Q2m(^) = ^ a»P2» + l(/*). 



Then 



2 f 1 



Un . . -o = 1 Q2»i(/a) P2n + l(At) <^U, 



(2m - 2™ - l)(2m + 2rc + 2) 

 and we obtain the expansion 



. (^.-s^ ^ywff . (16) 



„_ (2m — 2n — 1) (2m + 2n -h 2) 



which is convergent when fi is between + 1, both exclusive. 

 Similarly, with the same convergence, 



™~ + M)^ \ (2m^2n + l)(2m + 2n + 2) # * U '> 



The same expansions are valid for the "associated Legendre 

 functions " 



p«>), Q/00 = (i-V)™ /2 £f(p>» <W 



by operating on both sides. Thu< 



O r/ u \- y~ (4n + 3)P 2)l+1 ^) 



«*(/0-. V>*-2n-l)(2m + 2n + 2)' ' (18> 



lj*n + i . W - -* (2m _ 2n + 1)(2m + 2n + 2) • . (19) 



