328 BELL SYSTEM TECIIKICAL JOURNAL 



In this paper formulae for the alternating current resistance have 

 been worked out in detail with particular reference to the case of 

 relatively large conductors at high frequencies and to relati\ely 

 thin tubes. In general the auxiliary functions involved are expressed 

 as the product of the corresponding functions for solid wires by a 

 correction factor which formulates the greater generality due to 

 the variable inner boundary of the conductors. As far as possible 

 the symbols are the same as in the solid wire case but refer now to 

 the system of tubular conductors. Primes are added where the 

 letters denote the corresponding functions for the solid wire case. 

 This will hardly lead to confusion with the primes used in connection 

 with the Bessel functions to denote differentiation. 



The general solution is developed in section II. The alternating 

 current resistance of one of the tubular conductors is expressed as the 

 product of the alternating current resistance of the conductor with 

 concentric return and a factor which formulates the effect of the 

 proximity of the parallel return conductor. Section III is a sum- 

 mary of the general formula, special asymptotic forms and forms 

 for thin conductors. 



II. Ma IIIKMA IH AI. AXAI.VSIS AM) 1 )KKIVATI( >N OK FORMULAE 



We require the expression for the axial electric force, £:, in the 

 conductors. Since the tubular condiuior does not extend to r = 0, 

 the electric force must be expressed !>> the more general I'ourier- 

 Bessel expansion, 



£z= ^ An{Jn{p) + \„K„(p)\ cos ne, 



where 



p = />\/47rX/i/u) 



= ^ = xi\/i when r = a 



= i= yiy/i when r = a, 



a and a being the outer and inner radii, respectively, of the con- 

 ductors. The additional set of constants X„, Xi . . . Xn is to be deter- 

 mined b>' the conditions of continuity at the inner boundary of the 

 conductor. It is necessary to satisfy the boundary conditions at the 

 surface of one conductor only, since the symmetry of the system 

 insures that they will then be satisfied at the surface of the other also. 



