On the Criterion for Oscillatory Discharge of a Condenser. 149 . 



the necessary damping k= ,/ nA , or the logarithmic decrement 



per wave equal to ~ roughly. 



8UO0 ft J 



Putting p=2irn, where n is the frequency, this would give 



as the ratio of one amplitude to the next of the same sign 



n 



about e^oo. 



Since the decrease of L with maintained oscillations is due 

 to the concentration of current near the surface of the wire, 

 it is at once suggested that we must have, in the present case, 

 when the damping becomes important, an aum-concentration 

 of current. The following investigation, by the method of 

 Maxwell, shows that this is the case. In the discussion in 

 the ' Treatise/ vol. ii. Art. 689, Maxwell expresses the current- 

 density w at distance r from the axis of the wire by his 

 equation (3), which, modified by the introduction of /x (see 

 Lord Rayleigh, Phil. Mag. May 1886), reads 



- iTfiw = 1\ + 4 T 2 r 2 + 9 T 3 r 4 + . . . . + n 2 T n r 2n ~ 2 + .... 



The T's are functions of the time which are subsequently con- 

 nected with each other by the equations (10) (again modified 

 by the insertion of p), 



tt/. <ZT ir>» d»T 



1 p df" » p\n\f dt n ' 



where p is the resistivity. 



T is then expressed in terms of the total current C by 



cli. „ , clLi , 2 2 a p 



a dt=- C+ * /Xa dt ^ U aW +&C ' 



The reduction of these equations enables us to express w in 

 terms of C and its differentials with respect to the time, thus 



To apply this to damped oscillations, put 



the right-hand side of the equation last written then becomes 



e-^^pt [l+*w(i - 3) +«V(*V-p')(£ - \ ~ + ~) & 



