555, 556] Induction of Currents 485 



The energy of the currents is wholly kinetic so that we may take 



(515), 



.) .................. (516), 



as before ( 503). 



In the general dynamical problem, it will be remembered that T was a 

 quadratic function of the velocities. Thus i lt i 2 , ... must now be treated as 

 velocities and we must take as coordinates quantities x lt # 2 , , defined by 



Clearly x- measures the quantity of electricity which has flowed past any 

 point in circuit 1 since a given instant, and so on. Thus in terms of the 

 coordinates x lt x 2 , ... we have 



T=i(Ai*i 2 + 2A a A 1 A a +...) ..................... (517). 



There is no potential energy in the present system, but the system is 

 acted on by external forces, namely the electromotive forces in the batteries 

 and the reaction between the currents and the material of the circuits which 

 shews itself in the resistance of the circuits. We have therefore to evaluate 

 the generalised forces lt 2 , .... 



Consider a small change in the system in which x-^ is increased by 8^, so 

 that the current ^ flows for a time dt given by ^dt^Sxi. The work per- 

 formed by the battery is E^x^ the work performed by the reaction with the 

 matter of the circuit being equal and opposite to the heat generated in the 

 circuit, is R^dt. Thus if X l is the generalised force corresponding to the 

 coordinate x lt we have 



X-^Xi = EI$XI R^i^dt, 



so that X-i = E l R^ . 



The Lagrangian equation corresponding to the coordinate x l is 



a fdT\ dT _ 

 a*UJ~8^ 1J 



o 



or ^ (L^ + L l2 i 2 + ...) = E 1 - JB^, 



or again E l --^= R^ . 



The equations corresponding to the coordinates x. 2) x 3 , ... are 



Thus the Lagrangian equations are found to be exactly identical with the 

 equations of current-induction already obtained, shewing not only that the 

 phenomenon of induction is consistent with the hypothesis that the whole 

 mechanism is a dynamical system, but also that this phenomenon follows as 



