294 Prof. Thomson on the Mutual Attraction or Repulsion 



Hence, if D and E denote the quantities of electricity on the two 

 spheres in the present case, and if W denote the mechanical 

 value of the distribution of electricity on them, we have 



W=i(Dt^ + Ei;). 



Now if the two spheres, kept insulated, be pushed towards one 

 another, so as to diminish the distance between their centres 

 from c to c—dc, the quantity of work that will have to be spent 

 will be F . dc, since F denotes the repulsive force against which 

 this relative motion is affected. But the mechanical value of 

 the distribution in the altered circumstances must be increased 

 by an amount equal to the work spent in producing no other 

 effect but this alteration. Hence ¥ .dc= — «?W, and therefore 



^-~2 do ^^^)' 



where u and v are to be considered as varying with c, and D and 

 E as constants. Now, according to the notation expressed in 

 (13), we have 





(17). 



du dv 



Determining -y- and -r- by the differentiation of these equations, 



and using the results in (16), we find 



This expression agrees perfectly with (/), given above ; since, 

 by differentiating the equations {b) and (c) with reference to 

 c, we find that the quantities denoted above by S'j, S'^ S'g, &c., 

 P'l, P'g, P'g, &c., Q'l, Q'g, Q'g, &c., and expressed by the equa- 

 tions Q) and {h), are equal respectively to 



IrfS, IdS^ ldSs_ n 1 ^ 1 ^ l^^Pg « 



2 dc' 2 dc ' 2 dc ' ^^'' 2 dc' 2 dc' 2 dc* ' 



IdQ, IdQ^ IdQ 

 2 dc' 2 dc ' 2 dc 



monstration is founded. It was first published, I believe, by Helmholtz 

 in 1847, in his treatise " Ueber die Erhaltung der Kraft," by the trans- 

 lation of which in the last number of the New Scientific Memoirs a great 

 benefit has been conferred on the British scientific pubhc. 



