314 Models of the Aether. 



It is evident that, when the system is considered from the 

 point of view of general dynamics, the electric currents must be 

 regarded as generalized velocities, and the quantities 



(L 1 i 1 + Z, 2 i 2 ) and (Z 12 ^ + L 9 i 2 ) 



as momenta. The electromagnetic ponderomotive force on the 

 rings tending to increase any coordinate x is dT/dv. In the 

 analogous hydrodynamical system, the fluid velocity corresponds 

 to the magnetic force: and therefore the circulation through 

 each ring (which is defined to be the integral fvds, taken round 

 a path linked once with the ring) corresponds kinematically to 

 the electric current ; and the flux of fluid through each ring 

 corresponds to the number of lines of magnetic force which 

 pass through the aperture of the ring. But in the hydro- 

 dynamical problem the circulations play the part of generalized 

 momenta ; while the fluxes of fluid through the rings play the 

 part of generalized velocities. The kinetic energy may indeed 

 be expressed in the form 



where KI, c 2 , denote the circulations (so that KI and c 2 are 

 proportional respectively to ^ and 4), and NI, N n , N 2 , depend 

 on the positions of the rings ; but this is the Hamiltonian (as 

 opposed to the Lagrangian) form of the energy-function,* and 

 the ponderomotive force on the rings tending to increase 

 any coordinate x is - dK/dx. Since dK/dx is equal to dT/dx, 

 we see that the ponderomotive forces on the rings in any 

 position in the hydrodynamical system are equal, but opposite, 

 to the ponderomotive forces on the rings in the electric 

 system. 



The reason for the difference between the two cases may 

 readily be understood. The rings cannot cut through the lines 

 of magnetic force in the one system, but they can cut through 

 the stream-lines in the other : consequently the flux of fluid 

 through the rings is not invariable when the rings are moved, the 

 invariants in the hydrodynamical system being the circulations. 



* Cf. Whittaker, Analytical Dynamics, 109. 



