192 Dr. E. W. Hobson. On a Type of Spherical [Jan. 16, 



V 2 J 



2 



m + n + 1 m—n 2 



In + m\ 



2 n( n ~ m ~ 1 )n(-j) 



/m + w + 2 m — n + 1 \ 



F i— — i— .*^;» 



the upper or lower sign is to be taken in the exponential, according 

 as fi is above or below the real axis ; 



Q„» W = f .2- >■ '," ' . V -iW?±g±?, 2L=_" x _,\ 



^ W vrv 2 III l_ m \ V 2 2 ' 2 / 



where a = — e"^""*)**', & = e (»-»)*»*, if /t is above the real axis; 

 and a = e<3»» +»)*«, b = e < 8m+w >* ,rt , if a is below the real axis. 



Special conventions are made as to the values to be attached to the 

 functions at points in the cross-cut between the points .+ 1 ; these 

 conventions are so made that the values of the functions shall be real 

 for real values of m and n ; they are given by 



V n m (cos 6) — e * mm ¥ n m (cos 6 + . t) = e-i m ™ P n «- (cos 6 — . l) , 



e*' Q/*(cos6>) = £{e-£ m7rt Q/*(cos 6 + . i) +e* mjrt Q„ m (cos 6— . <)}; 



it is further shown that 



g-%w7» Q/* (cos 0+0. *) — e* m,rt (cos 0—0. £ ) = — ive mm P«™(cos 0). 



The following expressions are obtained for the f auctions in series 

 of powers of fi— V /x 2 — 1 : — 



