Electric Waves round a Large Sphere. 527 



Employing the asymptotic expansion for the zonal harmonic, 



— r^ is found to be of order mh cos ( m#— T ), and therefore 



the typical term contains 



cosech /3 X cosh*/3, . . . . (31) 



•or e^~&, which is ultimately evanescent, however great r 

 may be. This compensates any power of z present, and the 

 sum of the harmonics of order n much greater than z cannot 

 contribute to the value of the magnetic force. 



Formulce of summation. 

 It has been shown by the writer* that the series 



s =i>0 exp -^©< • • • • < 32) 



where z is great, and where differentiation with respect to 

 n/z = x does not increase the order of u or v in z, is equivalent 

 ■to the integral 



,("\-(v.+ 5 + 5+„W. • • (33) 



where n/z in the functions has been replaced by a continuous 

 variable x and where 



Wj, n 2 + l = ~(ei, e 2 ), 

 uv 



V = M^r = 



2 sin 



K (34) 



where 



so that 



^ s— L5©* ^-ljw^iw' [ 



(35) 



the accents denoting differentiations with respect to x. 

 If i/ has no zero or very small value at any point in the 

 range an asymptotic value for the sum may be obtained by a 

 series of integrations by parts. In this case, the result 

 becomes, to the second significant order 



S=-*** 



{5 + K^-'¥)}- ■ » 



* Mess, of Math. Oct. 1907. 



