Wave Propagation over Parallel Wires. 615 



Then, omitting the subscript in ly the axial electric force at 

 the surface of wire #1 is 



Ao(J (f) +Mi(f) cos 6 + /* 2 J 2 (f) cos 20+ . . . ), (15) 



and the value of the tangential magnetic force H d at the 

 surface of the wire is by (10) : 



-^° ( J '(f ) + Mi'Cf ) cos + W 2 '(f J cos 20 + . . . ). (16) 



Since 4-7T times the total current I flowing in the wire is 

 equal to the line integral of the magnetic force H# around 

 the circumference of the wire, it follows at once that 



&$& A = 21, (17) 



/JLip V 



which determines the fundamental coefficient A in terms of 

 the current in the wire. 



The resistance R of the wire per unit length is con- 

 veniently defined as the mean dissipation per unit length, 

 divided by the mean square current. The dissipation W in 

 the wire is very conveniently and simply formulated by 

 Poynting's theory of the energy-flow in the electromagnetic 

 field, which, applied to the present problem, gives 



W = ~(k-K,d0, (18) 



where E^ and H$ are the values at the surface of the wire, 

 as given by (15) and (16). If these series are substituted 

 for E z and E. e in (18) and the value of A is taken from (17), 

 •and if the resulting expressions are realized, it follows without 

 difficulty that 



R = R {l + l/22|M 2 ^^},. v (19) 

 u a + iv n = J n (£) = J n {bi^i), 

 u,/ + iv n ' = -jr3n(bi ^i). 



R denotes the a.c. resistance of the wire when the co- 

 efficients hi . . . h n are all zero ; that is, R is the resista?ice 

 of the wire where the return wire is concentric, which is 

 calculable from well-known formula and tables. The 

 functions u n and v m it will be observed, correspond precisely 

 with the welt-known ber and bei functions, which are 



2 S2 



