221] ELECTROMAGNETISM. 431 



given. For k = - 1 we get a resolution into forces proposed by 

 Weber and C. Neumann, and for k = one implicitly suggested by 

 Maxwell. Let us examine the case k = 1. 



(.6) - W-IjJ [ 



JlJ 



. COB 



2 r 



1 dr dr , , 

 i2 - 5- o~ ds^Sz. 

 JiJirfad* 



From this we obtain 



*w r r f f f ! 8r 3r s ! 8r 88r X 9r 8 

 (i 7) - $W = ^/s M- 5- r- Sr - - 5- = --- 5- o~ 

 J i J 2 IT 3*i 9s 2 r ds 2 dSi r d* 2 9 2 



Integrating by parts, the second term around the circuit 1, and 

 the third around the circuit 2, the integrated parts vanishing in 

 both cases, 



T t f f 1 8r 8r 8 A 



1 / 2 -^-^ ^- + ^ - 

 J i J 2 1^ 2 3*i 3s 2 3! V^ 



(18) 



T f [ ( I dr dr 2 9 2 r 

 = Ij/2 -^ -- - r- ^ + - 



J iJz ( r z ds!ds 2 r ds 



Since the integrand contains the factor e>r, work is done only when 

 the distances apart of some of the pairs of elements are changed, and 

 we may resolve the action into attractions between dsi arid ds 2 of 

 the magnitude 



drdr 9 2 r 



x 

 = /i/2 (2 cos (c?5j, ds. 2 ) 3 cos (r, ds^ cos (r, c?5 2 )}. 



This form for the elementary forces was given by Ampere*. Accord- 

 ing to this form, we see that parallel elements perpendicular to the 

 line joining them attract each other with a force 



Parallel elements having the direction of the line joining them 

 repel each other with a force 



* Ampere. "M6moire sur la thSorie math6matique des phenomenes Slectro- 

 dynamiques." Mem de VAcad. T. vi., 1823. 



