Pkofessok Thomson on Transient Electric Currents. 285 



forces (for instance, twice the equivalent of chemical action in the 

 batteries, should the electro-motive forces be chemical,) over and above 

 that which they would have bad to spend in the same time if the con- 

 ductors had been at rest merely to keep up the currents, because the 

 electro-dynamic induction produced by the motion will augment the 

 currents; while on the other hand, if the motion be such as to require 

 the expenditure of work against electro-dynamic forces to produce it, there 

 will be twice as much work saved off the action of the electro-motive 

 forces by currents being diminished during the motion. Hence the aggre- 

 gate mechanical value of the currents in the two conductors, when brought 

 to rest will be increased in the one case by an amount equal to the work 

 done by mutual electro-dynamic forces in the motion, and will be dimi- 

 nished by the corresponding amount in the other case. The same con- 

 siderations are applicable to relative motions of two portions of the same 

 linear conductor (supposed perfectly flexible). Hence it is concluded 

 that the mechanical value of a current of given strength in a linear 

 conductor of any form, is determined by calculating the amount of work 

 against electro-dynamic forces, required to double it upon itself, while a 

 current of constant strength is sustained in it. The mathematical 

 problem thus presented leads to an expression for the required mechanical 

 value consisting of two factors, of which one is determined according to 

 the form and dimensions of the line of the conductor in any case, irrespec- 

 tively of its section, and the other is the square of the strength of the 

 current. If it be found necessary to take inertia into account, it will be 

 necessary to add to this expression a term consisting of two factors, of 

 which one is directly proportional to the length of the conductor, and 

 inversely proportional to the area of its section, and the other is the 

 square of the strength of the current, to obtain the complete mechanical 

 value of the electrical motion. 



XXXVI. — On Transient Electric Currents. By Prof. Wm. Thomson. 



The object of this communication is to determine the motion of elec- 

 tricity at any instant after an electrified conductor of given capacity, 

 is put in connection with the earth by means of a wire or other linear 

 conductorof given form and given resisting power. The solution is founded 

 on the equation of energy (corresponding precisely to " the equation of 

 vis-viva" in ordinary dynamics) which is suflBcient for the solution of every 

 mechanical problem, involving only one variable element to be determined 

 in terms of the time. That there is only one such variable in the 

 present case follows from two assumptions which are made regarding the 

 data, namely, 



(1.) That the electrical capacity of the first mentioned, or principal con- 

 ductor, as it will be called, is so great in comparison with that of the 

 second or discharger, as to allow no appreciable proportion of its original 

 charge to be contained in the discharger at any instant of the discharge, 



