196 On a Type of Spherical Harmonics, [Jan. 16, 



The. following expressions are obtained : — 

 V p 2 — 1 cos (0— ty±iu)} n 



j,i=i n^T»j 



where u < log mod. \/{(/t + l)/(/t — l)} ; this expansion holds for 

 unrestricted values of n, m having all positive integral values ; 



{/*+ Vtf— lcos (0— *±m)}» 



__ ^ n(n) f p ^ ^ _2 e _ nm gin njr _ ^ 1 eTOt(0 -*-^), 



n(w+???) l 7r j 



where w>logmod. \/{(^+ 1 )/(/ 4 '~ 1 )}» w * s unrestricted and m has 



the values n— 1, 2 



The following generalisations of the well-known expressions of 

 Dirichlet and Mehler for P w (cos 6) are obtained : — 



P.- (cos 6) = J - sin- e f ^n+iH ^ 



2«n(— ^)n(m— J) J o (2cos0— 2cos0)*- OT 



where the real part of m + J is positive, and n is unrestricted. 



(,cos - 2?Mn (-l) n (m-l) 1 J, (2 cos 0-2 cos 0)*-™ rf0 



f 00 e -(»+i)t> "j 



-fcos (w+i— ?n)7r 7- — — -ti— ? > 



v 2 y J o (2cos/iv + 2cos0)»- ,}l J 



which holds, provided the real parts of m-f^, n—m + 1 are positive. 



Various other definite integral formulae are deduced which hold 

 under special conditions. 



The following recurrent relations are proved for unrestricted values 

 of n and m : — 



0.'-l) ar °^ W = (»-m+l)P4,( / «)-(»+l) / .P,»O t ), 



<y _l) = »/*P.» W-(^+m) PA W, 



(Sn+l)/iP.« Ot) -(»'*-«» + l)Pd?i + PA O) = 0, 



P^ +2 (/i) + 2(m + l)-^^^P„' w+1 W-(n-m) (?i + m + l)P„«(/0 = 0, 



V yu, 2 — 1 



with precisely corresponding formulae for the Q functions. 



The memoir concludes with an examination of the ring functions, 

 and the harmonics for the cone and the bowl ; in particular, con- 

 vergent series are obtained for both the tesseral toroidal functions. 



P, 



