1896.] Harmonics of unrestricted Degree, Orders $c. 193 



. N sm(n + m)7r II (n + m) , „ ■ , . 



cosnr n(w+^)n(— -j) 



F(£-fm, n+m + l,n+- 



?) 



jj (n -L) / 1 



F (i + m,w + »i + l 



where z denotes ya+ yV~— 1- 



A second expression for each of the functions is obtained in the 

 form of an integral, which might serve as an alternative definition of 

 the functions — 



1 IlCm— T\z+,z- 1 -) fon+m 



47rsm (n + m)7r 



the meanings of the integrands being as before precisely denned ; 

 these expressions are not readily deducible from the former ones. 



Expansions in powers of ^^J f ^ — i are obtained for the two f anc- 

 tions, and the following special cases are deduced : — 



p„» (cos,) = ! t;^) 



Au(» + J)i- (2sin0)* 2.2w + 3 



/ — — 3tt m?r\ / 5tt m7r\ "| 



cos [u-\- — -+— 2 A 2 . , , cos[n + f0 



(2 sin 0)f + 2.4.2n + 3.2w + 5 (2 sin 0)i J 



This series represents the function for unrestricted values of n 

 and m, provided sr/6<0<5?r/6; it is, however, shown that if n and m 

 are real, and such that n + m— 1, |-f m are positive, a finite number 

 of terms of the series represents the function approximately when 

 9 is not subject to the restriction, 



