Forced Vibrations of Electromagnetic Systems. 493 



the field there will be positive, and, by continuity, also posi- 

 tive outside the cylinder. Similarly, H must be negative at 

 any distance within which E is decreasing. We conclude 

 therefore that the filament solutions (372) ouly express the 

 settling down to the final state, and are not comprehensive 

 enough to be employed as fundamental solutions. 



69. Sudden starting of e longitudinal in a Cylinder. — In 

 order to fully clear up what is left doubtful in the last para- 

 graph, I have investigated the case of a cylinder of e compre- 

 hensively. The following contains the leading points. We 

 have to make four independent investigations : viz. to find 

 (1) the initial inward wave ; (2) the initial outward wave ; 

 (3) the inside solution after the recoil ; (4) the outside solu- 

 tion ditto. We may indeed express the whole by a definite 

 integral, but there does not seem to be much use in doing so, 

 as there will be all the labour of finding out its solutions, and 

 they are what we now obtain from the differential equations. 



Let Ej and E 2 be the E's of the inward and outward waves. 

 Their equations are 



E^-iaftqWUe, .... (377) 



E 2 =-(a/2?)WU a '«j .... (378) 



where U and W are given by (309), the accent means differ- 

 entiation to r, and the suffix indicates the value at r = a. To 

 prove these, it is sufficient to observe that U and W involve 

 € qr ande~ qr respectively, so that (377) expresses an inward 

 and (378) an outward wave ; and further that, by (310), we 

 have 



E 1 —E 2 = g at r = a, always; . . . (379) 



which is the sole boundary condition at the surface of curl of e. 

 Expanding (377), we get 



E 1=i 0^ ( R + S)[l + ?-y + ^I 



y W 



3 2 . 5.7.9 



*»* 



+ 



]«, • (380) 



where R+ S is given by (309). Now e being zero before and 

 constant after £ = 0, effect the integrations indicated by the 

 inverse powers of p, and then turn t to ^, where 



rti = vt + r — a. 



