HELMHOLTZ ON THE CONSERVATION OF FORCE. 141 



The equilibrium of electricity in a conductor cannot take place 

 until the resultants of the whole of the attractive forces of its 

 own electricity, and such other electrified bodies as may happen 

 to be present, are perpendicular to its surface ; because, were it 

 otherwise, the electric particles must be moved along the surface. 

 Consequently the surface of an electrified conductor is itself a 

 surface of equilibrium, and the vis viva gained by an infinitely 

 small electric particle in its passage from one conductor to 

 another is constant. Let C^ denote the vis viva gained by the 

 unit of positive electricity in its passage from the surface of the 

 conductor A to an infinite distance, so that Ca is positive for 

 positive charges; Aa the potential of the same quantity of elec- 

 tricity in regard to A when it occupies a certain point upon the 

 surface of A ; A^ the same in regard to B ; W„ the potential of A 

 upon itself; W^ the same of B ; V that of A upon B, and Q„ the 

 quantity of electricity in A, Qj in B ; the vis viva gained by the 

 particle e in its passage from an infinite distance to the surface 



-eC„=e(A„ + A,). 



If instead of e we set successively all the electric particles of the 

 surface A, and for A^ and Aj the corresponding potentials, and 

 add all, we obtain 



In like manner, for the conductor B, 



The constant C must not only possess the same value for one 

 and the same conductor, but also for separate conductors, if the 

 latter, when connected together in a manner by which the dis- 

 tribution of their electricities is not sensibly changed, exchange 

 no electricity with each other; that is, it must possess the same 

 value for all conductors possessing the same free tension. As 

 unit of measure for the free tension of an electrified body, we can 

 make use of a quantity of electricity which, distributed over a 

 sphere of the radius 1, placed beyond the distance where induc- 

 tion can take place, is in electric equilibrium with the said 

 body. If the electricity be distributed uniformly over the sphere, 

 the exterior action, as is known, will be the same as if the elec- 

 tricity was concentrated at its centre. Denoting the mass of 



I 



