m 



c = 



n! 



Un becomes 



(2.1)/ 



n(n— 1) ^ n(n— l)(n— 2)0i — 3) 



Un = .r" ^ .r"-2 + -^^ ^-^ ' .t-"-4 — . . . etc. . (5) 



2(2/^ — 1) ^ 2 4.(2«— l)(2w— 3) ' 



the well known form (but tbr a constant factor) of the zonal har- 

 monic function and, according to (4) : 



+1 +1 



Jn!n! C 

 (2n)/J ^ 



{x- -- 1/' dx = - 



2»+\n!y 



— 1 



— 1 



(2?«+l)/(2/0/ 



Putlinff 6'= , we tind, if by P„ the commonly used form of 



zonal harmonics is denoted, 



(2n)/ 



2".nhif 



from which 



+1 



—1 

 If the limits are + go and — oo it is rational to choose for (f„ : 



ffn = Ce-^'' 



Un = C 



dx" 



Putting 



C = 



(-!)« 



Un assumes the form : 



U,, = Un'e^ ^'=^6- 



-^'\ .r" 



v{7l — 1 ) 



~2\rr 



2» 



.."-^ -I 



„(„_l)(„_2)(n— 3) 



..)i - 4 



)i — \ 

 (-1)^ 



n. 



n-l . 



2«-i ! 



2 



2\2/ 

 ,'/; (y/ uneven) 



(Ö) 



or 



n/ 



(— 1)" {n even) 



w 



2»—/ 

 2 



(-1)"/^ 



