Electric Waves round a Large Sphere. 535 



are the second and last. Neglecting therefore the first and 

 third, as having sums of a lower order of magnitude, 



w =rffl)S 1 '*WS (58) 



since v 4 = v 2 when ^ n is ignored. 



Thus if v denote v 2 or v± so that zv = <f> n — <f> nr — m0, 

 then at the zero point, after some reductions, 



x = sin 0/(1 — 2c cos (j>-\- c 2 )i, 

 v = — c _1 (l — 2ccos0 + c 2 )% 

 v "= (1 — 2c cos + c 2 )f/(cos — c)(l — ccos0), 



and is positive, for cos is never less than cos 0, and there- 

 fore a fortiori not less than c. 



R»R nP = (1 — 2ccos0 + c 2 )/(cos0 — c)(l — ccos0), 

 whence 



u = 4^ sin 0/(cos0 — c)*(l — ccos0)*, 



and by use of the summation formulae, r " being positive, 

 and on reduction 



+ *r(l - 2c cos + c 2 )*}, (59) 



or 7/9 = — 4fcsin#(2£/a7r)£-^, .... (60) 



provided that 



r _f* 0sm 0^0exp.— t{i7T — kr(l — 2c cos 0-1- c 2 )*} ,«-. 

 ~J " v/2(cos0-cos^)l(l-2ccos0-rc 2 )i ' l ] 



The important elements of this integral are those near the 

 upper limit. Writing therefore 



= 0—u, 

 and retaining no powers of u higher than the first, 

 cos = cos -f u sin 0, 



R 2 



1 — 2c cos + c 2 = ~y — 2c u sin 0, 



and 



sin 



-iff) - 



taexp.— a( J7r-f&R — ka-^usind) 



(2u sin 0)1 



to the most significant order, by the usual theory of integrals 

 of this type. 



