Self-induction of Wires. 339 



say V , C , functions of &, by 



~LC Q w)dz 



f'(sv «-: 



Jo 



(110) 



L dp\w 7 Jo 



provided there be no energy initially in the terminal arrange- 

 ments. If there be, we must make corresponding additions 

 to the numerator, without changing the denominator of A. 

 The expression to be used for ujw is, by (106) and (107), 



l - = —tan (mz + e), (Ill) 



w bp v n 



remembering that m is a function of p. There are four com- 

 ponents in the denominator of (110), as there are three elec- 

 trical systems ; viz. the terminal arrangements, which can 

 only receive energy from the " line," and the line itself, which 

 can receive or part with energy at both ends. 



In a similar manner, if we make It, S, and L any single- 

 valued functions of z, subject to the elementary relations of 

 (71), Part IL, or 



-g =EC + LC, -g = SV, . , (112) 

 getting this characteristic equation of C, 



and, putting w for C and p for -j- } this equation for the 

 current function, 



A( s -J) =(R+LpV) . . . m 



and finding the u functions by the second of (112), giving 



-Sj™=g, (115) 



we see that the expansions of the initial states V and C can 

 be effected, subject to the terminal conditions (108). For 

 the normal potential and current functions will be perfectly 

 definite (singularities, of course, to receive special attention), 

 given by (113) and (114), as each the sum of two indepen- 

 dent functions, and the terminal conditions will settle in what 

 ratio they must be taken. (109) and (110) will constitute 



