Intelligence and Miscellaneous Articles, 1 59 



Let us consider au electrified conductor having a charge q, and 

 suppose that a quantity of electricity dq escapes from the electrified 

 body and disappears into the air. When this electricity dq passes 

 from a level surface where the potential function has a value V, to 

 the infinitely near level surface where the potential function has 

 the value V-\-ch>, designating by du the infinitely small portion of 

 normal comprised between the two surfaces at the point considered, 



the repulsive force exerted on dq is — — dq. The elementary work 



of the repulsion, in passing from one level surface to the surface 

 infinitely near is — dYdq. Consequently, when the quantity of 

 electricity dq removes to infinity, the corresponding work has the 

 value V'clq, if we call V the potential function at the surface of the 

 conductor, or, what is the same thing, in the interior of this con- 

 ductor : this expression has already been given by M. Helmholtz. 



But in proportion as the loss of electricity is effected, the charge 

 on the conductor diminishes; it is the same with the potential 

 function. The potential function V is proportional to the charge 

 q of the conductor; we can putY'ssa^, a being a constant peculiar 

 to the conductor. The work necessary to repel to infinity the 

 quantity of electricity dq is aq dq. Consequently, calling the initial 

 charge of the conductor q , the work necessary for repelling all the 

 electricity of the body to an infinite distance has for its expression 



Jo 

 V denoting the initial value of the potential function on the con- 

 ductor. 



Therefore the work consumed by the repulsion of all the electri- 

 city to infinity is equal to the potential of the electricity ; or, in 

 other terms, the mechanical equivalent of the external discharge is 

 equal to the potential of the electricity. The result would evidently 

 be the same for a system of electrified conductors. 



The value of T is independent of the path pursued by the elec- 

 tricity which escapes from the conductors ; it is easy to recognize 

 that this value remains the same when two equal quantities of op- 

 posite electricities meet on their passage and recombine. 



Let us in fact suppose that two quantities of electricity, -\-m and 

 — m, recombine in a point M. to form neutral electricity ; let Y be 

 the potential function in this point. Suppose that at the point M 



the electricity +mis repelled by the force —m-y-; the portion of 



the work T necessary to remove -fmto infinity, starting from the 

 point M, is Ym. The quantity — m, on the contrary, situated at 

 the point M, is attracted ; the attractive force has the same value 

 as the repulsive force; and when — m removes from M to infinity, 

 the corresponding work is equal to the preceding and of the con- 

 trary sign. The value of T remains consequently the same, whether 

 the two electricities recede to infinity or recombination takes place 



