HELMHOLTZ ON THE CONSERVATION OF FORCE. 123 



those between r = R and r=oo as those already consumed; the 

 former can immediately act, the latter can only be called into 

 action by an equivalent loss of vis viva. It is the reverse with 

 repulsive forces. If the points are situate at the distance R, as 

 the distance becomes greater vis viva will be gained, and the still 

 existing tensions are those between r=^R and r^oo, those lost 

 are between r = and r=R. 



To carry our law through in quite a general manner, let us 

 suppose any number whatever of material points with the masses 

 nil, ^2J ^39 ^^' denoted generally by m^ ; let the components of 

 the forces which act upon these parallel to the axes be X„, Y„, Z^, 

 the components of the velocities along the same axes u^, v^, w^, 

 the tangential velocity q^ ; let the distance between m^ and nif, be 

 7'ad) the central force between both being c^a^. For the single 

 point nin we have, analogous to equation (1), 



V vf/ \(l>anl dVn 



Y„=SL(y„-2,„)-J=m„^, 



where the sign of summation X includes all members which are 

 obtained by putting in the place of the index a the separate in- 

 dices 1, 2, 3, &c., with the exception of w. 



Multiplying the first equation by dx^ =u^dt, the second by 



dyn—Vndt, the third by dZn=Wndt, and supposing the three equa- 

 tions thus obtained to be formed for every single point of m*, as 

 it is already done for m„ ; adding all together, we obtain 



2[(z.-z,)rfz,g] =2gm,rf(c«,\)]. 



The members of the left-hand series will be obtained by placing 

 instead of a all the single indices 1, 2, 3, &c., and in each case 



