86 Dr. J. W. Nicholson on the Effective 



and possesses the properties 



^ w»K„(w)==— w"K„_i(w), 



K„ +1 (?/•) — K w _i(«)= — K„(w), 



K, +1 0/) + K /i . 1 00 = ~2 



du 



> 



^K o0 0=-K lW . 



(28) 



The analysis required in determining the corrected function 

 is very similar to that above, and leads finally to the function 



B ()fy)= |l- 



cos 2 a(l — 5 sin 2 a) 



liy\K {thp). . (29) 



Determination of Inductance and Resistance. 



The terms of the Fourier series involving <£ do not con- 

 tribute to the total current across a section of the wire. Let 

 ~Ete %pt be the mean value across a section, of the line integral 

 of impressed electric force per unit length perpendicular to 

 the section, and parallel to the central curve. If tsre tpt be the 

 total current, and (L, R) the effective self-induction and 

 resistance per unit length of wire, 



(Lip + K)w=E. 



(30) 



Now at this point, the problem becomes identical in process 

 with that of the straight wire, provided that the appropriate 

 functions A (kp) and B (ihp) are used for the wire and di- 

 electric respectively, in place of J (kp) and T£ (ihp) for the 

 straight wire. Quoting therefore the result in the latter case, 

 as developed in a former paper *, namely, if r be the radius of 

 the wire, 



E _ 2 flip f Jo (Jcr ) Ic K (ihr) \ 

 *r~ kr t JVO^J fi/a Ko'(Ur) J 



we find for the helix 



E 2fiip f A (kr) 



*r~ kr lA '{kr) 



k B (ilir)} 

 phi B '(ihr) ) ' 



■ (31) 



(32) 



* Phil. Mag. Feb. 1909, p. 259. The result requires a factor -1/a. 



