1896.] Harmonics of unrestricted Degree, Order, fyc. 195 



where n is unrestricted, and m must be such that the real part of 

 + h is positive, that of /t must also be positive ; under the same 



m 



restrictions 



P n "'(/ ( )--e--"^siu™7r . Qn"W 



n(n + m) (/i 3 — l)* Ml f ir sin 2 " 1 ^ 



also 



v } n(w) v y sin v T 7 



/* cos ^ 



= 4 ; ^7 r .* 



2^o { « + a/u 2 — l cos (0 -tt m) } w 



where ?>i is a real integer, w is unrestricted, and u is any real positive 

 quantity less than ^ log. mod. and the real part of is positive, 



0-*» fl n(n-m) n(m-i) (a * 1} 



J" (> + V 1 COsh w -)-n-m-i 2m lt . fi w 



where the real parts of m + |-, n — m + 1 must be positive ; 



Qn w (/t) = e™"*^ — cos (V-l)-* 5 " 



v 7 ^- — lcosh w) n+m sinh ~ 2iJl ^ 



where w? = -J log. mod.-^-i-^, and the leal parts of w + m-f-1, J—w 

 must be positive ; 



Qn-M = ^#- ( ^ r — — 



n(w-w) J (^4. y/ff-l cosh 

 where the real parts of n+m+l and » — ?n-|-l must be positive ; 



„ , . Il(n + m) pogev/C^ + D/^-i)] 



y ft m(^) = . e ^t_- — J . y^ — 1 cosh u}» cosh m«rf«, 



where the real part of n + 1 must be positive. 



