608 Mr. J. R. Carson on 



valuable researches of such mathematicians as Mie * and 

 Nicholson |. 



In the present paper the analysis of the problem starts- 

 with Maxwell's equations, but one simplifying assumption 

 is introduced ab initio — namely, that the exponential 

 propagation factor is a small quantity. The approximations 

 involved in this assumption are fully justified in all physical 

 systems which could actually be employed for the trans- 

 mission of electrical energy ; so that from a practical 

 standpoint the restriction thus imposed on the generality 

 of tie solution is purely formal. By aid of this simplifying 

 assumption the determination of the current distribution 

 in the wires is essentially reduced to a two-dimensional 

 problem, which is solvable from the boundary conditions 

 satisfied by the tangential magnetic force and the normal 

 magnetic induction at the surfaces of the wires. With the 

 current distribution in the wires thus determined, the expo- 

 nential propagation factor <y is solvable by applying the law 

 curlE=— /jud/dtil to an appropriate surface bounded by 

 a contour which includes line elements in the surfaces of 

 the wires. By this means it is shown that the propagation 

 factor satisfies an equation of the form 



7 2 /?>K = 2Z + ipL, 



where K is the electrostatic capacity between wires, Z the 

 "impedance" of the wire per unit length, and L the in- 

 ductance corresponding to the magnetic flux between the 

 wires. This equation is of exactly the same form as that 

 derivable from the telegraph equation, but differs therefrom 

 in that Z and L are both functions of the frequency f\2ir 

 and the parameter k (ratio radius of wire to interaxial 

 separation between wires). As formulated in the present 

 paper, the actual calculation of Z and L involves only the 

 computation of Bessel functions. 



The method of solution sketched above and worked out 



* G. Mie, Amialen cler Plujsik, vol. ii. pp. 201-249 (1900). In this 

 paper the problem is attacked in a fundamental manner. The results 

 arrived at are, however, limited to a restricted range of frequencies 

 and the parameter k (ratio radius of wire to interaxial separation). 

 Furthermore, Mie's method of attack does not admit of extension to 

 other types of transmission systems in which the surfaces of the 

 conductors are generated by lines parallel to the axis of transmission. 



t J. W. Nicholson, Phil! Mag. vol. xvii. p. 255 (1909), and vol xviii. 

 p. 417(1909). In these papers formulas are derived for the resistance 

 and reactance of parallel wires which are valid for a very wide range of 

 frequencies, but which are applicable only when the ratio of the radius 

 of the wire to the interaxial separation between wires is a relatively 

 small quantity. 



