522 Dr. J. W. Nicholson on the Bending of 



waves only, and convergent at all points on or outside the 

 sphere, 



y P = - sin 2 r (gW J«(te) -D^K^r)^ . (10) 



Reduction ofyp. 



Let the oscillator be now placed on the surface of the 

 sphere, so that r x = a. 



If new functions be introduced, defined by 



^)=(±y{^JZ}^ ■ ■ ■ (n> 



in terms of the Bessel functions of order m or n + -^, then 

 from the relation defining K m (s) it appears that 



£«»K.(*)-(^) l (iB > ,»-0«--*-S • • • (12> 

 the accent denoting differentiation with respect to 2*. Thus 



**K,(*) / /^^K m (^) = ^- 1 R w cos X n e*n y 



where 



tan X n=-iR/ (13) 



Similarly, with some reduction, 



t; z\ J m {z) = ( -g- V cos (<£„ + % n ) sec X». . (14) 



In all further work, z will denote &a, and thus E n , <t> n are 

 functions of argument ka. The corresponding functions of 

 kr will be denoted by R nr , <£ n ,- In accordance with this, 



.?»(«)= -^("2* )« '**> • • • (15> 



after some reduction. 

 Thus 



t . 2n $ 2m/7rR„ 2R n ,\idP„ 



X [sin <£„-*COS {<f>n + Xny Xn y-"l>nn 



and therefore 

 IP = ^ S" mtfyy K«*^+ «*-*'+*•) ^ . (16> 



