Thin Wires for Rapid!// Alternating Currents. 427 



given by Mr. Mathews and myself in our treatise on Bessel 

 Functions, for the resistance of a straight metallic wire carrying 

 a very rapidly alternating current. The equation we give is 



»'=v/ 



fjunlR 



where R is the resistance of a wire of length I and permea- 

 bility /jl to steady currents, and R' is its virtual resistance to 

 currents of frequency n/'Iir. Taking ^ as unity, r as the 

 radius of the wire, and k as its conductivity, Mr. Boynton 

 transforms this equation to 



*'=Ww 



which leads to the result that for n = 500,000 and £ = '0006, 

 R' = 36,000rR. 



This result, as Mr. Boynton says, is startling, and is not 

 confirmed by his experiments. 



I desire to point out that the transformed equation obtained 

 by Mr. Boynton is not correct, and should stand 



R' 



=R, V /^, 



which, with n and k as stated, gives 



R' = 21-7rR, 



For wire of 1 millimetre diameter — the case considered by 

 Mr. Boynton, this becomes 



R'=1-085R, 



so that the virtual increase of resistance in this case works out 

 to only 8*5 per cent. This number, however, is not a close 

 approximation to the true value of R'. Obviously, if we were 

 to make the radius of the wire about 8 per cent, less than half 

 a millimetre, 21*7rR would become R, and further diminution 

 of r would give R ; <R, results which are clearly erroneous, 

 To find a fair approximation recourse must be had to the 

 series by which, for a given n and a given r, the virtual 

 resistance can be calculated. Thus I obtain, from three 

 terms of the series, R'= 1*291 R, which, as the next term is 

 positive, must be somewhat too small. 



The series is really one of ascending powers of the product 

 /juVr^k 2 ; and it is only when this quantity is increased without 

 limit that we obtain the limiting formula stated above. In 

 other words, the formula gives the value of R' for a given 

 wire to a given degree of approximation only when n is made 

 sufficiently great ; and it cannot be applied to a case in which, 



