150 On the Criterion for Oscillatory Discharge of a Condenser. 



If we put this into the shape 



Ae-* p 'cos(/>« + /3), - 



A will be a quasi-amplitude ,of the disturbance at distance r 

 from the axis. Its value works out 



i + i^(i- ^) +«v [*y g - ^ + £) + ip-g - -J)] fo 



Now if kp = while 7? is finite, so that there are undamped 

 oscillations, then the last term in the last bracket is the first 

 (besides the initial unit term) which does not vanish; and 

 we see that its value falls off as r decreases — showing the 

 ordinary surface concentration. If, on the other hand, the 

 damping is so great as to make the second term of the series 

 more important than those which follow, then the value of A 

 will increase as r decreases from a — showing ^m-concentration. 



The coefficient of kpafi changes sign at r= — =. Assuming 



v z 



the preponderance of this term, this means that in the neigh- 

 bourhood of this value we pass from a greater value on the 

 inner parts to a less value on the outer, than would correspond 

 to a uniform distribution throughout the wire. 



From general reasoning it seems clear that if we think of 

 a rapidly damped disturbance propagated into a wire from its 

 boundary ; and if the alternations are slow enough to allow 

 the currents to penetrate to the core, we should expect to 

 find an axial concentration during the latter stages of the 

 phenomenon. 



It may be of interest in this connexion to mention another 

 case in which alternation and decay act in o])j)Osite ways as 

 regards inductance. If oscillatory currents are being kept 

 up in a primary coil, it is well known that the presence of a 

 secondary decreases the effective inductance. But suppose 

 that we have a steady current existing in a primary, that we 

 cut off the applied E.M.F., and allow the current to die away. 

 Then the nearness of a secondary coil will cause it to decay 

 less rapidly, which corresponds to an increase of the apparent 

 inductance. The case is worked out in Prof. J. J. Thomson's 

 ' Elements of Electricity and Magnetism,' p. 391. It is there 

 shown that the presence of a secondary of resistance S and 

 inductance N changes the exponential which expresses the 

 decay of current in the primary from 



to e 



