340 Mr. 0. Heaviside on the 



the solution ; except as regards the initial energy beyond the 

 terminals. 



It is, however, remarkable, that we can often, perhaps uni- 

 versally, find the expression for the part of the numerator of 

 (110) to be added for the terminal arrangements, except as 

 regards arbitrary multipliers, from the mere form of the Z 

 functions, without knowing in detail what electrical combina- 

 tions they represent. This is to be done by first decomposing 

 the expression for C 2 (dZ/dp) into the sum of squares, for 

 instance, 



C^=r 1 {f l (p)y + r 2 {Mp)\> + ..., . . (116) 



where r lf r 2 , . . . are constants. The terminal arbitraries are 

 then %Ajfi(p), %Kf 2 (p), &c. : calling these E 1? E 2 , . . . , the 

 additions to the numerator of (110) are 



-\K i r l f l (p)+K 2 r 2 Mp)+...\, . . (117) 



wherein the E's may have any values. This must be done 

 separately for each terminal arrangement. The matter is 

 best studied in the concrete application, which I may consider 

 under a separate heading. 



It is also remarkable that, as regards the obtaining of cor- 

 rect expansions of functions, there is no occasion to impose 

 upon R, S, and L the physical necessity of being positive 

 quantities, or real. This will be understandable by going 

 back to a finite number of variables, and then passing to 

 continuous functions. 



Let us now proceed to the far more difficult problems con- 

 nected with propagation along a dielectric tube bounded by 

 concentric conducting tubes, and examine how the preceding 

 results apply, and in what cases we can be sure of getting 

 correct solutions. Start with the general system, equations 

 (11) to (14), Part I., with the extension mentioned at the 

 commencement of Part II. from a solid to a tubular inner 

 conductor. Suppose that the initial state is of purely longi- 

 tudinal electric force, independent of z, so that the longitudinal 

 E and circular H are functions of r only. How can we secure 

 that they shall, in subsiding, remain functions of r only, so 

 that any short length is representative of the whole ? Since 

 E is to be longitudinal, there must be no longitudinal energy- 

 current, or it must be entirely radial. Therefore no energy 

 must be communicated to the system at z = or z=l, or leave 

 it at those places. This seems to be securable in only five 

 cases. Put infinitely conducting plates across the section at 

 either or both ends of the line. This will make V = there, 



