230 The Mathematical Electricians of the 



mutual energy of two current-elements containing charges e, e 

 respectively of each kind of electricity, is 





r 3 



If ds, ds' denote the lengths of the elements, and i, i f the currents 

 in them, we have 



ids = 2ev, i'ds' = 2V ; 



so the mutual energy of two current-elements is 



nf 



-(r.ds').(r.ds). 



The mutual energy of ids with all the other currents is therefore 



t(dt.a), 

 where a denotes a vector-potential 



By reasoning similar to Neumann's, it may be shown that the 

 electromotive force induced in ds by any alteration in the rest 

 of the field is 



-(ds.a); 



and thus a complete theory of induced currents may be 

 constructed. 



The necessity for induced currents may be inferred by 

 general reasoning from the first principles of Weber's theory. 

 When a circuit s moves in the field due to currents, the velocity 

 of the vitreous charges in s is, owing to the motion of s, not 

 equal and opposite to that of the resinous charges : this gives 

 rise to a difference in the forces acting on the vitreous and 

 resinous charges in s ; and hence the charges of opposite sign 

 separate from each other and move in opposite directions. 



The assumption that positive and negative charges move 

 with equal and opposite velocities relative to the matter of 



