Inductance of a Wire to an Oscillatory Discharge. 437 



and L" . , , , y 



-r^ =aA+ (<zfLS/p)» COS^ 



MHSWf ' ' ' <21> 



Discussion of the Results for High Frequencies. 

 On putting & = 0, in equations (23) and (24), to reduce to 



7T 



the case of sustained simple harmonic waves, 5=1, 0=--\ 



u 



whence, denoting by single dashes these special values of R" 

 and L", we obtain 



g- = Sfatp ; (25) 



"-'{*+</£} > • • • • <*> 



which are Lord Rayleigh's high-frequency formulae*. 



Referring again to equations (23) and (24), we see that 

 for a given value of _p, if k varies from to co , the factor 

 involving s increases without limit while that involving 

 increases to unity. Hence, with increasing damping, it 

 appears that R" and U f each increase also, while ever the 

 equations remain applicable. Now an infinite value of k 

 involves zero frequency f. And a certain large, though 

 finite, value of k would prevent the frequency being classed as 

 u high." 



Dividing equation (23) by (25) gives 



g'=(2 s 3 )icos| = K S a y (27) 



Thus, for a given value of k, the ratio W/Rf is independent 

 of the frequency of the waves. It is therefore convenient to 

 deal with K a function of k only, rather than with R"/R 

 which is a function of p also. 



Differentiating to k, we have 



1Tr ok cos 77 + sin -r 

 1k~ 711 ' (28) 



* Equations (26) and (27), p. 390, loc. cit. 



t This follows from the fact that electric currents or waves generated 

 by an oscillatory discharge may be represented by e~ kpt cos pt, in which 

 Jcp is finite, so k is infinite only when p is zero. 



Phil. Mag. S. 5. Vol. 47. No. 288. May 1899. 2 H 



