Theory of Electromagnetic Action. 445 



The quantities N b N 2 , &c. are generalized components of 

 electrokinetic momentum. 



Using Lagrange's equations of motion and introducing Lord 

 Rayleigh's Dissipation Function F, we have, since the coeffi- 

 cients in T do not contain the coordinates y, for the impressed 

 electromotive forces in the different circuits the typical 

 equation 



f* + ^=E, (4) 



dt T dy k k Ki 



But 



F = ±2R. 



kVk 



where R& is the resistance in the circuit indicated by the 

 suffix k. Hence 



^=B»-B* (5) 



The electromotive forces of the type dN k /dt are the parts of 

 the impressed electromotive forces which are employed in 

 working against the electromotive forces due to induction. 

 Thus —dBjJdt is the type of the actual electromotive force 

 due to induction. Since this may be written in the form 



_ddT_ 

 dt dy k 



we see that the inductive electromotive force tends to diminish 

 the electrokinetic energy. 



Again, the forces which must be applied to work against 

 the reacting internal electromagnetic forces are 



dT _ dT 

 dx x ' dx 



~^r> &c -> 



where x v x 2 , &c. are coordinates determining the geometrical 

 positions of the circuits. Thus, the mutual electromagnetic 

 forces having equal and opposite values to these, tend to 

 increase the electrokinetic energy. These two results must 

 be kept clearly in mind in what follows. 



This dynamical theory, if applicable to a system of ordinary 

 circuits, must, if Ampere's theory of magnetism be true, be 

 also applicable to a system of magnets, or a system composed 

 partly of ordinary circuits and partly of magnets ; otherwise 

 Ampere's theory cannot be a complete expression of the facts 

 of the case. Maxwell has applied it only to the case of two 

 circuits, and its applicability to the case of a circuit and a 



