( 802 ) 



or, for n even, from : 



1 rt, «i «« 



1 '- j ^ [-.... — — = 



n -\- I n — 1 n — 3 1 



n-\-2,^ n-^l^ n—l^ 3 



1 ^9 «a ^9 



2n — 1 ^ 2n—3 ^ 2n — b ^ n — 1 



for n odd, from : 



1 a, a. (1/1—2 



1 i 1 ' h . . . . -^^— = 



71 -}- 2 ?i n — 2 3 



1 a, a, ct/i— 2 



n -]- 4 ?i -\- 2 n 5 



2n — 1 2w — 3 2n — 5 w 



On eliminating successively from these equations a^ ,a^ . . . . or 

 a ,((2 ... . we find for the general expression of the polynomium : 

 nCn—\) n(n-l)(n-2)(n-3) 



^" 2.(2«-l) ^ 2.4.(2n-l)(2n-3) ^^ 



i. e., but for a constant factoi', that of zonal liarmonics, which 

 we shall call P-functions. 



This might have been expected as the condition ^2), from which (4) 

 arises, holds good also for the P-functions. 



The (3-functions may, therefore, be considered as generalized 

 P-functions, the latter presenting a special case of the former; if 

 we write (2) : 



h I Qnic'^'dx — , 



then : 



hQn^P (5) 



if /:„ be defined so that : 



kn Q„ = 1 for .r = 1 . 



The nse of this constant no doubt offers advantages in treating 

 problems relating to the potential theory, but for our purpose it 

 would be of no importance and, in practice, entail superfluous work; 

 some expressions certainly take a simpler form by its use, but what 

 is thereby gained on the one hand is largely lost on the other as 



