224 The Mathematical Electricians of the 



Hence, the mutual potential energy of the two currents is 



- . dS 



which hy Stokes's transformation may be written in the form 



(ds.ds') 



This expression represents the amount of mechanical work 

 which must be performed against the electro-dynamic pondero- 

 motive forces, in order to separate the two circuits to an infinite 

 distance apart, when the current-strengths are maintained 

 unaltered. 



The above potential function has been obtained by con- 

 sidering the ponderomotive forces ; but it can now be connected 

 with Faraday's theory of induction of currents. For by 

 interpreting the expression 



(B . dS) 



If' 



in terms of lines of force, we see that the potential function 

 represents the product of i into the number of unit-lines of 

 magnetic force due to s' t which pass through the gap formed by 

 the circuit s ; and since by Faraday's law the currents induced 

 in s depend entirely on the variation in the number of these 

 lines, it is evident that the potential function supplies all that 

 is needed for the analytical treatment of the induced currents. 

 This was Neumann's discovery. 



The electromotive force induced in a circuit s by the motion 

 of other circuits s', carrying currents i' t is thus proportional to 

 the time-rate of variation of the potential 



(ds.ds'). 



so that if we denote by a the vector 



