Electric Waves round a Large Sphere. 165 



these higher approximations in a more extended investiga- 

 tion of the region of brightness. 



Before proceeding further at this point, it is necessary to 

 make some remarks about the sum of the series. Although 

 there is no group of terms of supreme importance on account 

 of a vanishing derivate of the function v, it might be thought 

 that since u rises in order in the neighbourhood of any pole 

 which it may have, terms in such a region might supply the 

 important part of the sum. But the poles of xi must arise 

 from the zeros of the function 



d/ds . *ft K m 0), 



none of which are real. Moreover, their imaginary parts 

 are all large, so that such terms will necessarily have an 

 exponential factor in their sum, of large real argument. 

 This argument cannot be positive from elementary physical 

 considerations, and therefore it is negative, and the sum of 

 any such set of terms decreases in a rapid exponential manner 

 round the surface, and may be ignored in comparison with 

 the terms considered in this section. The rigorous analysis 

 of this point will follow later, these rough indications being- 

 enough for the present. 



Since in the integral of (87) there is no zero point, and 

 no difficulty introduced by poles of the function u, an inte- 

 gration by parts is legitimate. This is taken between the 

 limits e and infinity, corresponding to ?z = and w=oo, and 

 e is in this case (2~) -1 . The term at infinity may be 

 neglected as corresponding to harmonics of infinite order, 

 which have been dealt with already, and for the term at 

 #=1/2;?, the value of « in (88) may be used. 



The order of the integral thus becomes that of 



u„jV* <, [lvJ] 6 . 



where zv e = +i>4>, and U 6 = u H = (2t) _1 , 



rejecting ~ -1 throughout. The integral thus has zero order 

 in z at most, and by an inspection of the operation G(0), yp 

 would therefore not be of greater order in linear magnitudes 

 than (ha 2 )~ l . When the sphere is absent, the corresponding 

 order of this quantity is known to be k, as found in the 

 investigation of the region of brightness. Accordingly, the 

 magnetic force when the sphere is present is at most of 

 order (ka)~ 2 relatively to its value when the sphere is absent, 

 and this is approximately 10 -12 in the numerical case typical 

 of wireless telegraphy. 



