I9I0.] LONG ELECTRIC WAVES ALONG WIRES. 367 



obtained from m = i to m = n — i. We thus get: 



V,= Z ^.sin—-.'-'. (12) 



A,n and pm are complex quantities. Writing />„,= />„/ -\- ipm", we get 

 from (6), (9) and (10) : 



p;= 4sm— ^nK'-RSJ (13) 



V 



LS' 4L''S' 



2C'2 



p"= Jr^, = /'. (14) 



p,n" is thus independent of ;;/. The real part of (12) may now be 

 written 



w — 1 klHTT 



V, = e-'"' Z ^,„ cos {p:t - <f>J sin ^^ . (15) 



where Am and </>», are new arbitrary real constants. 



The currents in the several coils may be obtained from (3) com- 

 bined with ( 12 ). Taking the real part we find 



C, =e-'"'' y, B cos (/ 't - ylr) cos (2 /^ - i) (16) 



where Bm and \\im are known in terms of Am and </>,„ in (15). 



The last four equations give the complete solution of the problem. 

 The constants Am and </>,„ may be determined by Fourier's method 

 when the initial conditions are known. 



Now let n increase indefinitely while R' , L' , S' and K' all decrease 

 indefinitely. Let L = limit L'li/l, and similarly for the others. Let 

 8a- be the distance between two coils, so that u8x^l. Measuring .r 

 from the end of the line corresponding to k^o, we have k^^nx/l 

 and we get in the limit : 



" HVKX 



V= e-r'"' Z ^„^ cos {p^t - <^ J sin ^- , (17) 



1 ^ 



C = e-P"' Z ^„. cos {p'j - y}rj cos -J- , (18) 



PROC. AMER. PHIL. SOC. XLIX. I97 Y, PRINTED JANUARY 21, I9II. 



