and Inductance of a Concentric Main, 525 



with the wave shape of the alternating currents. The " mea- 

 surement " of inductances by means of alternating currents 

 o£ unknown wave shape often leads to waste of time. Even 

 when the wave shape is similar to a sine curve, yet if we do 

 not know how the inductance will vary with the frequency 

 the results have only a very limited application. 



It would obviously be extremely useful to have formulae 

 which would take into account the appreciable variation of 

 the density of the current which occurs over the cross section 

 of the wires with high frequency currents. Even to fix 

 inferior and superior limits to the possible values of the 

 inductance would be a great help in many cases. Unfortu- 

 nately the mathematical difficulties in the way of arriving at 

 a solution in the case of a helical coil are very great. The 

 author, therefore, has made a study of the simplest problem 

 of all, namely that of a concentric main, as a preliminary to 

 attacking the more difficult problems. A study of this 

 problem is also of importance at the present time * in con- 

 nexion with the discussion that is taking place amongst 

 electrical engineers as to the magnitude of the skin losses in 

 power transmission cables. 



The problem was first discussed by Maxwell f. He obtains 

 a few of the terms of a series by means of which the effective 

 resistance of the inner conductor can be computed at low 

 frequencies. Apparently, however, he did not fully appreciate 

 the importance of the results given by the formula. In May % 

 1884, Oliver Heaviside discussed the " throttling " effect in a 

 core, that is, the increased resistance, the reduced inductance, 

 and the tendency to surface concentration. He uses two 

 functions M and N in his solution, which Kelvin subsequently 

 called the ber and bei functions. In January § 1885, he 

 described clearly the true nature of the current flow in a 

 wire, laying particular stress on the initial surface effects 

 and subsequent penetration. Lord Kayleigh ||, adopting 

 Maxwell's method, next discussed the problem and gave a 

 formula for the effective resistance of the inner conductor at 

 very high frequencies. Oliver Heaviside IF subsequently made 



* See the report of the evidence on the London Electric Power Bills 

 given before Sir Luke White's Committee in the House of Lords (Nov. 

 1908). 



t ' Electricity and Magnetism,' vol. ii. § 690. 



% ' The Electrician,' p. 583, May 3, 1884, or < Electrical Papers,' vol. i. 

 p. 353. 



§ ' The Electrician,' Jan. 3, 1885, or ' Electrical Papers,' vol. i. p. 429. 



|| Phil. Mag. xxii. pp. 381-394 (1886), or < Scientific Papers,' vol. ii. 

 p. 486. 



■ft ' Electrical Papers,' vol. ii. p. 64 et seq. 



