1 1 7-1 20] Mechanical forces 1 03 



119. As an illustration, we may consider the force between the two 

 charged spheres discussed in 116. 



dW 



The force tending to increase c, namely -^ , is 



" p 2 4. 2 F F + a F 

 c l '~8^ jM ' 2+ ~gc 2 ' 



and substituting the values 



Pii = - + terms in - , 







1 



#B = g- 



it is found that this force is 



,. 2 



- h terms in . 

 c c 



Thus, except for terms in c~ 4 , the force is the same as though the charges 

 were collected at the centres of the spheres. Indeed, it is easy to go a stage 

 further and prove that the result is true as far as c~ 4 . We shall, however, 

 reserve a full discussion of the question for a later Chapter. 



120. Let us write 



i ( Pn E^ + 2p l2 E,E 2 + ...)= W e , 

 Hfri^i a +2? u F 1 F, + ..0= W>. 



Then W e and W v are each equal to the electrical energy ^EV, so that 

 W e + W v -2EV=0 ........................... (50). 



In whatever way we change the values of 



EU E 2 , ..., Fj, F 2 , ..., fj, 2 > ..., 



equation (50) remains true. We may accordingly differentiate it, treating 

 the expression on the left as a function of all the E's, F's and 's. Denoting 

 the function on the left-hand of equation (50) by <f>, the result of differentia- 

 tion will be 



Now 8i_yFe 



-LI \J W ^\ T-f ^ ir-t 



~ ^ ' " " >' 



so that we are left with X 8^ = 0, 



