Electric Waves round a Large Sphere. 533 



to x 9 the sum of the series (50) is 



tv r*ao 

 yp = e 4 z I XJ e lzV dx, 



where 



iv' 

 x = (n + J)/*, and U = (W/2 sin £v')* 2 , . (52) 



e being 3/22, or zero to the order contemplated. 



This integral contains one, and only one point, in the range 

 at which v' = 0, and it is well known* that in such a case, 

 the important part of the integral is, if v " is positive as in 

 the present problem, 



(27r/zv ")*\Jo,oe tzv °-± l7r , .... (53) 



the zero suffix referring to the zero point. 



The ultimate value may be independently shown to be of a 

 higher order of magnitude in z than the greatest term of the 

 harmonic series for yp. Substituting therefore for the 

 integral, 



Now since v ' = 0, U 0)0 = w by (52). Thus writing 

 x = sin X in (50, 51), 



U 0)0 as -sin e x (2z sin d tan i soc(0 i -d)/a 2 ir)i , 



zv = — <p nr -f <pn — m0 = —zll/a = — / 1 1 , 



after some reduction, remembering that 



sin- 1 (c sin ^) = 0,-0 



and using the values of R», </>„, <j> nr given in (18, 40). 



Now , . , . , n 



r = sin l x — sin -1 ex — u, 



so that ,, /! *\-i /i 2 i\-i 



v = (1— ar) *— 0(1— <rar) ■ 



V = sec d x -c sec (^-(9) = T sec6>, sec^— 0) ? 

 and after final reduction 



yp = -2ik^ 2 sin 2 ^- t/lR , 



or the magnetic force is given by 



ya S -2^gj 9 in^- , * B (54) 



* (?/". a paper by Lord Kelvin, Roy, Soc, Proc. xlii. p. 80, and other 

 investigators. 



