235] INDUCTION OF CURRENTS. 



where the coefficients L, M have the form obtained in 220 if a 

 single homogeneous medium is present, and in any case may be 

 defined as magnetic fluxes as in 231. The electrokinetic 

 momentum cf any circuit, 



(3) p s =j^ = M ls I l + ...+L s ! s ...+M m I n) 



may be defined as the total flux of magnetic induction through 

 that circuit in the positive direction due to all the currents. We 

 have already found for any positional force, equation (i), 



_ 



The force P s belonging to any cyclic coordinate q s consists of the 

 impressed electromotive force E s , due to chemical, thermal, or 

 other action, and the dissipative term given by Joule's law, 

 R S I S , where R s is the resistance of the circuit. Accordingly 

 we have 



= %{MI 1+ ...+l.I....+MM 



If we write this in the form 



d v< 



&s~ 



(6) 



we see that the current in any circuit may be calculated by 

 Ohm's Law provided that we consider acting beside the electro- 

 motive force E s an additional electromotive force dp s /dt. This 

 is called the electromotive force of induction, and from the above 

 definition of p s we see that it is equal to the time-rate of diminu- 

 tion of the flux of magnetic induction through the circuit in the 

 positive direction. The law of induction was announced in virtually 

 this form by Faraday*, and was obtained from theoretical con- 

 siderations involving the idea of work by Neumann f, HelmholtzJ, 

 and Kelvin . The above equation is the general equation of an 



* Exp. Res. 114, 3082. 

 t Neumann, loc. cit. 



$ Helmholtz, Ueber die Erhaltung der Kraft. Berlin, 1847. Wiss. Abh. } Bd. i. 

 p. 12. 



B.A. Report, 1848. Math, and Phys. Papers, Vol. i. p. 91. 



