Self-induction of Wires. 425 



We see also, from (148), that if T be the magnetic energy 

 of any normal system per unit length of line, then 



2T = C 2 ^; (149) 



and therefore, if Q be the dissipativity in the conductors, 



Q = WC 2 -T = C 2 (ft"-p^\ . . . (150) 



Now consider the connection of the two solutions for the 

 normal functions. Since the equation of C in general is, 

 by (145), 



l(iS)= E "°-«' <«« 



the normal C function, say w, is to be got from 



d_ 



dz [ 



with d/dt=p in R" and S", making them functions of z and p. 

 Let X and Y be the two solutions, making 



w = X + qY, (153) 



where q is a constant. The normal V function, say u, is got 

 from w by the first of (145), giving 



if X'=dX/dz, Y=dY[dz. 



In X and Y, which together make up the w in (153), p has 

 the same value. Therefore, in (147), supposing d to be X 

 and C 2 to be Y, we have disappearance of the right member, 

 making 



t(VA-VA)=0, or VA-VsC.rr constant, 

 or XY'-YX' = S" x constant =7iS", say, . . (155) 



leading to the well-known equation 



connecting the two solutions of the class of equations (152); 

 which we see expresses the reciprocity of the mutual actioities 



