PROPAGATION OF PERIODIC CURRENTS 521 



Then (as shown in the next paragraph), for the particular differential 

 element which contains the point u, the quantities f(x)dx and 

 \^dF(x)/dxJ^dx must be replaced by — AF{u) and AF{u) respectively; 

 that is, 



f{u)du = — AF(u), ^ (65) 



^^ildu = AF(u). (66) 



Equations (65) and (66) can be obtained, by a limiting process, 

 from equation (2) of Appendix I, which for the present purpose will 

 be written in the form 



f(x)dx = -1 — dx -Tj— dx + fg{x)dx. (67) 



It will be recalled, from Appendix I, that this equation was derived 

 by applying the second curl law to a differential rectangle extending 

 from X to X ■{- dx; but x may equally well be a point within the 

 differential segment dx, and for the present purpose it will be so re- 

 garded. The limiting process now consists in letting dF(x)/dx ap- 

 proach infinity while dx approaches zero, but in such a way that the 

 product [_dF{x)ldx'}dx approaches a preassigned finite value, denoted 

 by AF{x). Then, in the limit, the last two terms on the right side 

 of (67) vanish so that (67) reduces to 



f(x)dx = — AF{x). 



Thus we obtain equations (65) and (66), where u denotes, for distinc- 

 tion, the particular value of x at which F{x) is discontinuous. 



Remarks on the Terminal Impedances and the Equivalent Electro- 

 motive Forces in Them 



The arbitrary terminal impedances Zo and Z« (Fig. 2) need not 

 actually be localized. They may, for instance, be the impedances 

 offered by other lines to which the given line 0-5 may be connected, 

 and these other lines may themselves be situated in arbitrary im- 

 pressed fields; in particular, 0-5 may be merely a segment, of any 

 length, forming part of a given line in an arbitrary impressed field. 



From this broad view, any ' equivalent electromotive forces' situated 

 in the terminal impedances Zq and Z, may advantageously be regarded 

 as being situated in the ends of the line itself (that is, in the end-points 

 X = and x = s), these electromotive forces being then regarded as 

 pertaining primarily to the line-segment 0-5 rather than to the terminal 



