490 Dynamical Theory of Currents [CH. xvi 



great, the currents tend to distribute themselves in such a way as to make 

 the kinetic energy of the currents a minimum subject only to the relations 

 imposed by Kirchhoff's first law. 



This result may be compared with that previously obtained ( 357) for 

 steady currents. We see that while the distribution of steady currents is 

 determined entirely by the resistance of the conductors, that of rapidly 

 alternating currents is, in the limit in which the frequency is infinite, 

 determined entirely by the coefficients of induction. 



562. As a consequence it follows that, in a continuous medium of any 

 kind, the distribution of rapidly alternating currents will depend only on the 

 geometrical relations of the medium, and not on its conducting properties. 

 In point of fact, we have already seen that the current tends to flow entirely 

 in the surface of the conductor ( 537). We now obtain the further result 

 that it will, in the limit, distribute itself in the same way over the surface 

 of this conductor, no matter in what way the specific resistance varies from 

 point to point of the surface. 



MECHANICAL FORCE ACTING ON A CIRCUIT. 



563. Let 6 be any geometrical coordinate, and let be the generalised 

 force tending to increase the coordinate 6, so that to keep the system of 

 circuits at rest we must suppose it acted on by an external force . Then 

 Lagrange's equation for the coordinate 6 is 



and therefore, since the system is in equilibrium, we must have 



If the energy of the system were wholly potential and of amount W, the 

 force would be given by 



Thus the mechanical forces acting are just the same as they would be if 

 the system had potential energy of amount T. 



564. Let us suppose that any geometrical displacement takes place, this 

 resulting in increases 80 lt S# 2 , ... in the geometrical coordinates O l} # 2 , ..., and 

 let the currents in the circuits remain unaltered, additional energy being 

 supplied by the batteries when needed. 



The increase in the kinetic energy of the system of currents is 



