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XIX. On the Extra Current. By Oliver Heaviside *. 



LET a wire possessing uniform electrical properties through- 

 out be of length I, resistance M, electric capacity cl, and 

 let its coefficient of self-induction be si. Further, let P and Q 

 denote the two ends of the wire, and x the distance of any 

 point from the end P. Let v be the electric potential at the 

 point x at the time t, and Q the quantity of electricity that has 



passed that point from the time t — 0, so that -~ is the current. 



The differential equation of the potential of the wire may be 

 found from the following two equations : — 



-§-- ■ « 



-t'*§**9- •;.-;■ •<* 



The first expresses the fact that the quantity of electricity 

 existing on the surface of the wire between sections at x and 

 x + 8x at any moment is the product of the potential and the 

 capacity of the portion of wire considered. The second ex- 

 presses that the electromotive force at the point x at any 

 moment is the sum of the electromotive force producing the 



current — and the rate of increase of the momentum of that 



current. By eliminating Q we obtain 



dm\~ Clc dt + CS dt 2 ' ••<<•< (3j 



If 5 = 0, the above equation becomes 

 d?v _ ,dv 



d^- ck Tt' 



the differential equation of the linear flow of heat, or of elec- 

 tricity in a submarine cable, the practical solution of which for 

 a wire of finite length can only be accomplished with the 

 assistance of Fourier's theorem. And if k = 0, we have 



d 2 v __ d 2 v 

 d^ 2 ~~ CS a¥ ; 



which is of the same form as the equation of motion of a vibra- 

 ting wire, the solution also requiring the use of Fourier's 

 theorem. It is therefore probable that the same method must 



* Communicated by the Author, 



