Forced Vibrations of Electromagnetic Systems. 435 



telligible solutions, which may be multiplied to any extent ; 

 and for the illustration of peculiarities of a striking character. 

 The case of vibratory impressed E.M.F. in a thin tube is very 

 rich in this respect, as will be seen later. At present I may 

 remark that the results of this paper have little application in 

 telegraphy or telephony, when we ^ are only concerned with 

 long waves. Short waves are, or may be, now in question, 

 demanding a somewhat different treatment*. We do, how- 

 ever, have very short waves in the discharge of condensers,, 

 and in vacuum-tube experiments, so that we are not so wholly 

 removed from practice as at first appears. But independently 

 of considerations of practical realization, I am strongly of 

 opinion that the study of very unrealizable problems may be 

 of use in forwarding the supply of one of the pressing wants 

 of the present time or near future, a practicable sether — 

 mechanically, electromagnetically, and perhaps also gravita- 

 tionally comprehensive. 



50. Mathematical Preliminary. — On account of some 

 peculiarities in BesseFs functions, which require us to change 

 the form of our equations to suit circumstances, it is desirable 

 to exhibit separately the purely mathematical part. This will 

 also considerably shorten and clarify what follows it. 



Let the axis of z be the axis of symmetry, and let r be the 

 distance of any point from it. Either the lines of E, electric 

 force, or of H, magnetic force, may be circular, centred on 

 the axis. For definiteness, choose H here. Then the lines of 

 E are either longitudinal, or parallel to the axis ; or there is, 

 in addition, a radial component of E, parallel to r. Thus the 

 tensor H of H, and the two components of E, say E longi- 

 tudinal and F radial, fully specify the field. Their connexions 

 are these special forms of equations (2) and (3): — 



* The waves here to be considered are essentially of the same nature 

 as those considered by J. J. Thomson, u On Electrical Oscillations in a 

 Cylindrical Conductor,*' Pro*. Math. Soc. vol. xvii., and in Parts I. and II. 

 of my yaper ''On the Self-induction of Wires," Phil. Mag-. August and 

 .September 1886; viz. a mixture of the plane and cylindrical. But the 

 peculiarities of the telegraphic problem make it practically a case of 

 plane waves as regards the dielectric, and cylindrical in the wires. The 

 " resonance '* effects described in my just-mentioned paper arise from the 

 to-and-fro reflexion of the plane waves in the dielectric, moving parallel 

 to the wire. This is also practically true in Prof. Lodge's recent experi- 

 ments, discharging a Leyden jar into a miniature telegraph-circuit. On 

 the other hand, most of such effects in the present paper depend upon the 

 cylindrical waves in the dielectric; and, in oixh-r to allow the dielectric 

 fair play for their development, the contaminating influence of diffusion 

 is done away with by using tubes only when there are conductors. In 

 Hertz's recent experiments the waves are of a very mixed character 

 indeed. 



