HELMHOLTZ ON THE CONSERVATION OF FORCE. 139 



The gain in vis viva caused by passing from the distance R to r is 



-/?'^='t 



__^/^-_?/?// 



When they pass from the distance ^ to r, the expression is 



— -^-^. Let us denote this last quantity, which is the sum of the 



tensions consumed by the motion from x to r and of the vis 

 viva produced^ in conformity with Gauss in his magnetical 

 researches as the potential of the two electric elements for the 

 distance r ; the increase of vis viva due to any motion whatever 

 is then equal to the excess of the potential at the end of the 

 route over its value at the beginning. 



CaUing the sum of the potentials of an electrical element, with 

 reference to the whole of the elements of an electrified body, 

 the potential of the element towards the body, and the sum of 

 the potentials of all the elements of an electrified body towards 

 all the elements of another, the potential of the two bodies, we 

 obtain again the gain in vis viva in the difference of the poten- 

 tials, provided that the distribution of the electricity in the bodies 

 is not changed, hence that the bodies are idio-electric. If the 

 distribution undergo a change, the magnitude of the electric 

 tensions in the body itself becomes altered, and the vis viva 

 gained must then be different. 



By all methods of electrifying, equal quantities of positive and 

 negative electricity are generated ; in the neutralization of the 

 electricities between two bodies, one of which, A, contains as 

 much positive electricity as the other, B, does of negative, half 

 of the positive electricity goes from A to B, and half the negative 

 from B to A. Let the potentials of the bodies upon themselves 

 be W„ and W^, their potential towards each other V, we then 

 find the entire vis viva which has been gained, when we subtract 

 the potentials of the moving electric masses on themselves and 

 towards each of the other masses, before transfer, from the same 

 potentials after the transfer has been effected. It is here to he 

 remarked that the potential of two masses changes its sign when 

 the sign of one of the masses is changed. Hence the following 

 potentials come under consideration : — 



