140 Mr. 0. Heaviside on the Extra Current. 



pose -=- of such magnitude that \J 4,iV jj — 1 is appreciably 



= 2i7r v/ -j, then the time of a complete oscillation, including 



a positive and a negative current at any point, is nearly 2 \/a/3, 

 so that there are \ / °j- complete oscillations in the time 2«. 



The strength of these oscillations is proportional to a /— ; so 



that the larger -~ the weaker the oscillations, they being at 



the same time more rapid in the same proportion. 

 The time-integral of the extra current is 



V* Yd (a? 2x 



kl 2 V 2 I + 



where the first part is the same at all points, and is due entirely 

 to the momentum of the initial current. The second part is 

 the excess of the positive over the negative currents due to the 

 initial charge, and is twice as great at the end P as at Q. This 

 is the same when s = 0, or when there is no self-induction. 

 The work done in the wire by the extra current is 



when -^ is the same at all points, and 



/t all points, ai 



when -rr varies with x. Hence the amount of work done by 



dt . . si /V\ 2 



the first part of the current in equation (16) is -^ x (y^ ) , and 



Y 2 cl . l ^ kU 



by the second part — T — , which was the energy of the initial 



charge =2] VcCl— j) dx. 



As another example, suppose that before the time i=0a 



V . . 



uniform current yj existed in the wire, with potential 



v = Y ll— r), and that at the time t = both ends of the wire 

 are instantaneously and simultaneously insulated without allow- 



