150 Mr. 0. Heaviside on Electromagnetic Wares, and the 



V=y,+W^-^) .... (84) 



C = C + 2^e^(/^), . . .-. (85) 



where the summations range over all the p roots of<£ = 0, 

 subject to (79); whilst u and w are the V and C functions in 

 a normal system expressed by 



w=cosm2, u — msm mz-±-(K + $p); . . (86) 



and Y , C are the final steady V and C. In the case of the 

 solitary root (82) we shall find 



-p^^Kl + n,), (87) 



but for all the rest 



dcf> I dm 2 . 2 ,. /00 . 



-?$ = 2(KTW)^¥ ( + ' h h ■ (88) 



Realizing (84), (85) by pairing terms belonging to the two 

 p's associated with one m 2 through (79), we shall find that 

 (QQ), (67) express the solutions, provided we make these 

 simple changes: — Divide the general term in both the summa- 

 tions by 



1 + n x cos 2 ml, 



and the term following C outside the summation in (67) by 

 (1 +r*i). Of course the m's have now different values, as per 

 (83), and V , C are different. 



14. There are several other cases in which similar reductions 

 are possible. Thus, we may have 



Z =n (R + Lp) + « , (K+ S/>)-\ 



Z, = n, (R + Lp) + V(K + Sp)" 1 , 



simultaneously, n , n ', n x , nj being any lengths. That is, 

 apparatus at either end consisting of a coil and a condenser in 

 sequence, the time-constant of the coil being L/R and that of 

 the condenser S/K. Or, the condenser may be in parallel 

 with the coil. In general we have, as an alternative form of 

 <jS>=0, equation (78), 



tan?n? = (Z + Z t )\(H + Lp)l\- 1 

 ml l-m 2 l 2 Z Z l \(R + Lp)l\- 2; 



(89) 



