Equations in a Homogeneous Isotropic Medium. 35 



coils are connected, with corresponding instantaneous changes 

 in their momenta. A loss of energy is involved. 



It is scarcely necessary to remark that the true physical 

 theory involves other considerations on account of the dielectric 

 not being infinitely elastive and on account of diffusion in 

 the wires ; so that we have sparking and very rapid vibrations 

 in the dielectric. The energy which is not wasted in the 

 sp.vrk, and which would go out to infinity were there no con- 

 ducang obstacles, is probably all washed practically in the 

 heat of conduction-currents in them. 



8. Impressed Forces. — Given initially Eq and Hq, we know 

 that the diverging parts must either remain constant or sub- 

 side, and are, in a manner, self-contained ; but the solenoidal 

 fields, which would give rise to waves, may be kept from 

 changing by means of impressed forces eo and ho- Thus let 

 Eq and Hq be solenoidal. To keep them steady we have, in 

 eqiiations (1), (2), to get rid of pE and jt?H. Thus 



curl (Hq— ho) = 47r^-Eo, 

 curl (eo— Eo) = 47r^Ho, 



} . . . . (14) 



are the equations of steady fields Eo and Hq, these being the 

 forces of the fluxes. Or 



curl ho = curl Hq— 47r^Eo, 

 curl eo = curl Eo + ^tt^Hq, 



} . . . (14a) 



gives the curls of the required impressed forces in terms of 

 the given fluxes, and any impressed forces having these curls 

 will suffice. 



Now, on the sudden removal of e^ ho, the forces Eq, Hq, 

 which had hitherto been the forces of the fluxes, become, in- 

 stantaneously, the forces of the field as well. That is, the 

 fluxes themselves do not change suddenly, except in such a 

 case as a tangential discontinuity in a flux produced at a sur- 

 face of curl of impressed force when, at the surface itself, the 

 mean value will be immediately assumed on removal of the 

 impressed force. We know, therefore, the effects due to 

 certain distributions of impressed force when we know the 

 result of lea\dng the corresponding fluxes to themselves with- 

 out impressed force. It is, however, the converse of this that 

 is practically useful, viz. to find the result of leaving the fluxes 

 without impressgd force by solving the problem of the esta- 

 blishment of the steady fluxes when the impressed forces are 

 suddenly started ; because this problem can, often be attacked 

 in a comparatively simple manner, requiring only investigation 

 of the appropriate functions to suit the surfaces of curl of the 



D2 



