Forced Vibrations of Electromagnetic Systems. 145 



we know that cooling a metal and packing the molecules 

 closer does increase its conductivity. But as they do not 

 form a compact mass in any substance, they must always 

 allow a partial transmission of electromagnetic waves in the 

 intervening dielectric medium, and this would lead to the 

 diffusion method of propagation. We do not, however, ac- 

 count in this way for the dissipation of energy, which requires 

 some special hypothesis. 



The diffusion of heat, too, which is, in Fourier's theory, 

 done by instantaneous action to infinite distances, cannot be 

 physically true, however insignificant may be the numerical 

 departures from the truth. What can it be but a process of 

 radiation, profoundly modified by the molecules of the body, 

 but still only transmissible at a finite speed? The very 

 remarkable fact that the more easily penetrable a body is to 

 magnetic induction the less easily it conducts heat, in general, 

 is at present a great difficulty in the way, though it may 

 perhaps turn out to be an illustration of electromagnetic 

 principles eventually. 



12. Infinite Series of Reflected Waves. Remarkable Iden- 

 tities. Realized Example. — When, in a plane-wave problem, 

 we confine ourselves to the region between two parallel planes, 

 we can express our solutions in Fourier series, constructed so 

 as to harmonize with the boundary conditions which repre- 

 sent the effect of the whole of the ignored regions beyond the 

 boundaries in modifying the phenomena occurring within the 

 limited region. Now the effect of the boundaries is usually 

 to produce reflected waves, Hence a solution in Fourier series 

 must usually be decomposable into an infinite series of separate 

 solutions, coming into existence one after the other in time if 

 the speed v be finite, or all in operation at once from the first 

 moment if the speed be made infinite (as in pure diffusion). 

 If the boundary conditions be of a simple nature, this decom- 

 position can sometimes be easily explicitly represented, indi- 

 cating remarkable identities, of which the following investi- 

 gation leads to one. We may either take the case of plane 

 waves in a conducting dielectric bounded by infinitely con- 

 ductive planes, making E = the boundary condition ; or, 

 similarly, by infinitely inductive planes producing H = at 

 them. But the most practical way, and the most easily fol- 

 lowed, is to put a pair of parallel wires in the dielectric, and 

 produce a real problem relating to a telegraph-circuit. 



Let A and B be its terminations at z = and z = l respec- 

 tively. Let them be short-circuited, producing the terminal 

 conditions V = at A and B in the absence of* impressed force 

 at either place. Now, the circuit being free from charge 



Phil. Mag. S. 5. Vol. 25. No. 153. Feb. 1888. L 



