of Force between Electric Currents. 453 



and an element ds f of the other circuit, and /j, is a constant 

 depending on the nature of the medium in which the shells 

 are placed. 



3. If we were here to assume that the potential of mutual 

 action between each pair of elementary currents is of the same 

 form, 



V =fiii' ds ds f , 



r 



we should of course obtain the correct value for the potential 

 of two closed circuits ; and therefore this assumption would 

 satisfy all the results obtained from experiments with closed 

 circuits. This form of mutual potential was proposed by 

 F. E. Neumann, but seems to have been abandoned, because it 

 would not satisfy the results obtained, or supposed to have been 

 obtained, in experiments with open currents. Now, according 

 to the views of Maxwell, no such thing can exist within the 

 range of our experiments as an unclosed current, because the 

 current, if not closed by conductors, closes itself by means of 

 change of displacement in the dielectric. If this be true, no 

 experiments can ever lead to results inconsistent with the 

 above simple law of potential. Further, even without assu- 

 ming the truth of Maxwell's theory, it appears to us that the 

 experiments which were supposed to be inconsistent with the 

 above law admit of interpretation consistent with it, as we 

 hope to show. 



4. It was believed, however, to be established by experi- 

 ment, that the attractive force exerted by any closed circuit 

 upon any element of another current is always normal to the 

 element (canons III. and VI.). According to the above 

 law of F. E. Neumann, the impressed force exerted by the 

 closed circuit on an isolated element would not necessarily be 

 normal to it. It was thought necessary, therefore, to invent a 

 law of force between two elementary currents which should 

 satisfy this supposed experimental result, and at the same time 

 should give the correct value for the potential of two closed 

 circuits. 



According to F. E. Neumann's law, the force between two 

 elements of given strength depends only on their distance and 

 the angle, e, which their directions make with one another ; 

 it is independent of the angles 6 and 6' which their directions 

 make with r, the line joining them. But, by a known geo- 

 metrical theorem, 



for any two closed curves in space. Hence, if we assume for 



