528 



BELL SYSTEM TECHNICAL JOURNAL 



where An , B,/, C'n and Dn are expressible in terms of An, Bn, Cn and 

 Dn and of the currents, electrical constants and dimensions of the 

 system. In the neighborhood of ^2 = a, F is given by a similar ex- 

 pression in the coordinates ^2, di. The application of the boundary 

 relations at ri = a and ri = a then, as explained above, leads to a set 

 of equations which determine the arbitrary constants in terms of the 

 currents, electrical constants and dimensions of the system. When 

 these equations are solved and the arbitrary constants are known, 

 equation (24) becomes 



C2 



ZJy + Z„J2 = 2 Zi + 2MtC0 log- + ^s]h 



+ llliiiu^Xogj^+lAl., (34) 



where 



A Ji + A,„/2 



.?. -^' ^" 





C2 



COS^ ] {An - Cn) 



+ (sin'^)(5„ - Z)„)j 



The formal solution is then complete, Aj/i + ^mli representing the 

 correction in the series voltage drop of the primary circuit due to the 

 proximity effect. 



Solution by Successive Approximations 



As the set of simultaneous equations, upon which depends the de- 

 termination of the arbitrary constants An, Bn, Cn and Dn, involves 

 an infinite number of unknowns, a direct solution is, in general, im- 

 possible. Consequently, some method of successive approximation is 

 required. The convergence of the harmonic sequences indicates the 

 practicability of the following procedure in the present problem. 



(1) Determine first approximations .4 „^°^ and Bn^^^ by boundary con- 

 ditions at ri = a, neglecting the summations in C„ and D„. For the 

 first approximation only ^/"^ and ^i^") will be required and the series 

 may be represented by their leading terms. 



(2) Determine C„(') and Dn^'^ in terms of ^,/») and S„(»^ by con- 

 ditions at ^2 = a. 



