194 Dr. E. W. Hobson. On a Jype of Spherical [Jan. 16, 



n , ^ /- n^ + ^o l ^t^^j + f ) l 2 -4m 2 



cos ^^0+^+^ 



(2 sin 0)1 + • • • 



tlie same remarks holding" for this series as for the last one, 



Q m , ~i t \ /— n(n-\-m) e f l 2 — 4m 

 re m (cosh *) = e mm v/tt -) ^ / , — , , * <( 1 



n(w+|) (2smht)U 2.2» + 



4m" e" 



3 2 sinh ^ 



_j_l 2 — 4m 2 . 3 2 — 4 m 2 e~ 2 * \ 

 2.4.2rc + 3.« + 5 (2 sinh *) 2 ""J 



where cosh mnst be greater than 3/2 v /2. 



The following asymptotic values of the functions are obtained : — 



n O) . p.wfoosg^ = A / 2 [/ 1— 2m 2 \ . / 77 m~ 



1 — 4m 



cot 



/ . 7T m7r\] 



8» V ' 2 '4 2 



1— 4m 



+ — - — cot sin ?i + ^6> + - + — 



n t , -*> P„»(cosh *) = — - e ""*_ Ji-l 1 I 



n(») y,,,^ yi_ e -2* l s» » *» 1— e -2 */' 



n(w— m) V w ^i_ e -2* L 8?* % 



_1— 4r/r 1 1 

 4^ 1 — e - J 



The following expressions for the functions involving definite 

 integrals along real paths are obtained ; these include many known 

 formulas as special cases : — 



2 



*V*00-" - e~ m * 1 sin mir . Q/'O) 



Yl(n +m ) (ir — l)* m pr. — . .■*•"'„■, 



