12 i HELMHOLTZ ON THE CONSERVATION OF FORCE. 



for b also all the values of b, which are greater or smaller than a 

 already possesses. The sums divide themselves therefore into 

 two portions, in one of which a is always greater than b, and in 

 the other always smaller, and it is clear that for every member 

 of the one portion 



a member 



rpq 



must appear in the other portion : adding both together, we 

 obtain 



-{^p-x,)(,h^-dx,)^: 



rpq 



drawing the sums thus together, adding all three and setting 

 \d^{xa-.x,f + {y^-y,Y^ (Za-z,y^ =r«,c/r„„ 



we obtain 



--S\j>a,dr^'\ = ^\^mj{q\)\, (3) 



or 



-2[^j'"Va»<^'-»»]=2[i™,Q\]-2[^'m,g\], . (4) 



where R and Q, as well as r and q, denote contemporaneous 

 values. 



We have here at the left-hand side again the sum of the 

 tensions consumed, on the right the vis viva of the entire system, 

 and we can now express the law as follows : — In all cases of the 

 motion of free material points under the influence of their at- 

 tractive and repulsive forces, whose intensity depends solely upon 

 distance, the loss in tension is always equal to the gain in vis 

 viva, and the gain in the former equal to the loss in the latter. 

 Hence the sum of the existing tensions and vires vivce is always 

 constant. In this most general form we can distinguish our law 

 as the principle of the conservation of force. 



In the deduction of the law as given above, nothing is changed 

 if a number of the points, which we will denote generally by the 



