﻿52 Dr. J. G. Leathern '.Effect of a Long 



Annalen der Phi/sik, 1809, vol. lxvii. p. 255. The result is, 

 for equation (7), 



v 2 v 2 iog(pa-/7rv 2 a 2 ) ' v J 



in which j» 2 /^ 2 * s relatively so small that it may be neglected. 

 Prof. Thomson's result is 



»«=*;+,• , rt**' . . . .(80 



v 2 log (pa/lirv 2 y 2 ) 



where log y = *5772 157 ... . Now as the complete quantity 

 whose logarithm is taken is very small and therefore has a 

 large logarithm, the difference between the denominators in 

 formulae (8) and (8 X ) is small compared with either, so the 

 one formula is as good an approximation for m 2 as the other. 



High frequency currents. 

 When wais great y(ina)/g' (in a) =i, and equation (6) becomes 



(i/ea) 2 log (i/ca) 2 =— iaamp/27rv 2 ; . . . (9) 



for frequencies sensibly less than that of Bose's most rapid 

 vibrator the right side of this equation is small. The approxi- 

 mate solution is 



Prof. Thomson's result differs from this only by the presence 

 of an additional factor IM7 2 in the argument of the logarithm ; 

 this implies the addition of a relatively very small term to 

 the denominator, so that the two approximations are equally 

 good. 



The case of a wire with a return circuit in the shape of a 

 coaxial cylinder may be treated by the same method. It is 

 unnecessary to discuss it here, for the reasoning is the same 

 as that given by Prof. Thomson in the Uncyclopcedia Britannica 

 and in ' Recent Researches/ The only difference is that the g 

 and h functions are to be used instead of Bessel's functions, and 

 the approximate values are to be taken from Table (2). The 

 final results are found to differ from Prof. Thomson's only by 

 the absence of relatively very small terms. 



The approximate distribution of currents in the wire, and 

 the restriction of high frequency currents to a certain 

 thin " skin," can be discussed as easily by the present mathe- 

 matical formulation as by the use of Bessel's functions. 



The present note makes no claim to be regarded as a com- 

 plete account of the particular problems in the transmission 



m 



