526 Dr. A. Russell on the Effective Resistance 



most important contributions to our knowledge of the subject. 

 He was the first, for example, to give the approximate formula* 

 for the effective resistance of a hollow cylindrical conductor 

 carrying a low frequency current. This formula is a par- 

 ticular case of the general formula given in this paper. He 

 also gives general descriptions of how the current-density 

 varies in the conductors. He omits so many steps, however, 

 in some places that it is very laborious to follow his reasoning. 

 Three years afterwards Lord Kelvin f gave a practical 

 solution for the effective resistance of a solid inner conductor. 

 It is virtually the same as that given by Heaviside. He 

 gave a table, which we shall examine later, of the numerical 

 values in important practical cases. Sir Joseph Thomson J 

 also gives practical formula? for the effective resistance and 

 inductance of a concentric main having a solid inner conductor 

 when traversed by very high frequency currents. 



The complete solution given in this paper is obtained from 

 elementary electrical considerations, a knowledge of Ohm's 

 law and of Faraday's law of induction being all that is 

 assumed. The author proves the mathematical formulas at 

 length, as most of them are new and some of them will be 

 helpful in other physical problems. It will also enable any 

 slips he may have made in the algebraical work to be readily 

 detected and easily rectified. He shows, however, that from 

 his solutions all the previous solutions can be readily deduced, 

 and as most of them are complex functions of the electrical 

 and geometrical data of the main, the errors, if any, must be 

 very minor ones. 



II. Mathematical FoKMULiE. 

 1. The Differential Equation. 

 The differential equation § to which our problem leads is 

 B 2 s , 1 Jdi ___ ra 2 cM ,j^ 



where m and o) are constants and i is a periodic function. If 

 we assume that i varies according to the harmonic law, and 

 that its frequency is o>/27r, we may write i = ue U)i \ where u is 

 a function of r but not of t, and i stands for \/ — 1. 



* L. c. ante, p. 192, formula (72). 



f Journ. of the Inst. of. El. Eng\ vol. xviii. p. 4 (1889) or Math, and 

 Phys. Papers, vol. iii. p. 491. 



X i Recent Researches/ p. 295. * 



§ This equation was first discussed by Joseph Fourier [Memoires de 

 lAcademie, Tome iv. (for the year 1819)]. 



