Resistance and Inductance of a Helical Coil. 85 



Writing therefore 



A T / n COS 2 a 



where u is a new variable, only required in its first term, not 

 involving (ka)~ 2 , then on substitution, and rejection of (lea)-' 2 

 in the result, 



(d* , 1 d J A 8 cos 2 * rf <*J , / 9 o 2 n/^ 2 , 1 d , A rfJ 



I =-s H =- -f-ll ?«= — t- # -I- 21 + (2 — 6 cos 2 a) ( y-=- 4- — -= hi) x 7 ~. 



\<fo 2 x dx J x dx dx v y V^^ 2 « dx J dx 



By properties of the Bessel function of zero order, this 

 becomes 



hence 



& + ^i +1 >= +4(1 - 5sinSa > J ^ 



ignoring the complementary function. Finally, 



u=-i(l-5sin 2 *)x 2 J (x),. . . . (25) 



again ignoring a complementary function. The total form 

 thus ignored is A^J^.i') 4- B J (#), of which the first term 

 pertains to the redundant solution of the original equation. 

 The second has no relation to the original J (.r) with unity as 

 coefficient. We write therefore 



A^{l-2*4g**4*yW . . (26) 



as the value of A to the second order. Substitution in the 

 differential equation directly verifies this solution, and it 

 therefore corresponds to the function J (x) of the unbent 

 wire. 



Solution for the Dielectric. 



The above values hold for the interior of the wire of resis- 

 tivity cr, provided that k 2 = — 47r/zt/>/°"> where pj^tr is the 

 frequency of oscillation. For the dielectric, the appropriate 

 function outside a straight wire would be K (thp), where 

 h=p/V takes the place of k. The function is defined by 



-T 



K n (iu)=\ d<f> cosh n(f>e- mc09h( > . . . (27) 



