260 Dr. J. W. Nicholson on the 



This may also be written 



K n (i,7+^)=J (r)K n (ic)+t (-*)• J. (r){K B+t (w) +K„_ s (*c)} 



s - 1 ... (24) 



Now the series 



CO 



K ?l ( Jzc)J"o(7w) + 2 (— i) s {K n+s (i7ic) +K n - s (ilic}J s (Jir) cossO 



is a solution of the equation (9) which, when 6 = 0, takes 

 the form of the right-hand side of (24). Such a solution is 

 also cos n<f> . K n (Ji/)), where p and cf) are connected with r and 6 

 as in the figure. 



Thus p 2 = r 2 + c 2 + 2cr cos6. .... (25) 



j. rsin# 



tan<£=— - -5, (26) 



^ c + r cos 6 v ' 



and when = 0, cos n<f>K n (t7ip) = K n \i7i(c + r)}, 



Thus the two solutions only differ by a function vanishing 

 with 0, and therefore capable of expression in a series of 

 sines. This function is at once seen to be zero, by an appli- 

 cation of Fourier's theorem. The function cos ??<£K n (fc/ip) and 

 the series last written do not change sign with 0. Thus if 

 (p, </>) are connected with (r, 6) as before, 



cos ncf) . ~K n (i7ip) 



= K n (ihc)J {7ir) +I(- o) s {K n+s (L7ic) + K n - s {L7ic)}J s (7ir)cossd (27) 



and similarly 



cos nO . K n (c7ir) =K n (die) J (Jip) 



+ 1 t s {K n+S (ttic) + K n -s (die) } J s (Tip) cos s<£. (28) 



s=l 



Effect of t7ie Return Wire. 



It has been shown that if an impressed force E (ignoring 

 a time-factor e 1 ^) acts per unit length in a single wire of 

 radius a, the vector potential produced outside is 



H = BK (^r), 



where 7i=p/C, C being the velocity of propagation of electro- 

 magnetic disturbances in the medium, and 



B= ^ J ' (*a)/(K ' (cha) J (Tea) j. J ' (Tea) K (i7ia)), 



