98 M. E. Edlund on the Nature of Electricity . 



Subtracting nov/ the sum of the last two numbers from that 

 of the first two, wx obtain as the expression of the action which 

 two current-elements exert upon one another when they move 

 in opposite directions in parallel circuits : — 



^:^[l-^cos2^J, .... (11) 



which is found to be in full accordance with Ampere's empiric 

 formula. 



What has just been said refers to the supposition that the ve- 

 locity h is the same in both circuits. It is easy, however, to 

 prove that the above demonstration applies equally to the case 

 in which the velocity is greater in one circuit than in the other. 

 Let us suppose that the velocity in the circuit a' V (fig. 3) is A', 

 while that in the circuit ab is equal to h, that h!<h, and that 

 the motion is in the same direction in both circuits, viz. towards 

 h and V. It is evident that the relative velocity is not altered 

 by the absolute velocity of both molecules being increased or di- 

 minished by an equal quantity. If each of the molecules m and 

 m^ receive a velocity // in a direction opposite to the preceding, 

 the molecule m' wall come to rest, w^hile m will continue to move 

 in the same direction as before, but with the velocity h — h^. 

 Consequently their relative velocity, according to the preceding 

 considerations, will be h — h' cos 6. We obtain then, for the 

 action no. 1, 



-7?' [i-a{h-h') co^e-^ikih-h'fco^^e+ikih-hyii-cos^e)] 



for the action no. 2, 



+ ^ [l + ah' cos e-iklJ^ cos'- e-rWt'^ (I- cos^ 6)]. 

 No. 3 gives 



777W 



and, lastly, no. 4, 



mirJ 



Subtracting the sura of the last two results from that of the 

 first two gives : — 



which, as is seen, agrees with experiment. 



