and Conductance Operators. 485 



quency ; yet, as it is only true for impulses that the com- 

 bination behaves as a coil of inductance L , it is better to 

 signify this fact in the name, to avoid confusion. This will 

 be specially useful in the more general case in which both 

 energies are concerned. 



Secondly, let the system be electrostatic. Then, in a similar 

 way, we may write the electric energy in the form 



U=|S V 2 , (19) 



in terms of the V at the terminals, where S is a function of 

 the real permittances and of the resistances. S is the 

 impulsive permittance of the combination. It is also the 

 sinusoidal S' at zero frequency. 



In (18) L is positive, and in (19) S is positive. The 

 momentum or electromotive impulse at the terminals in the 

 former case is L 0, and in the latter case is — S RV, where 

 E, is the steady resistance. The true analogue of momentum, 

 however, is charge, or time-integral of current, and this, at 

 the terminals, is — S Y, corresponding to L C. 



6. Passing to the general case, and connecting with the 

 resistance-operator, let T be the current at the terminals at 

 time t when varying, so that 



Y = Zr=(Z +pZ ' + ifZ " + ...)V, . . (20) 



where the accents denote differentiations to p, and the zero 

 suffixes indicate that the values when p = are taken. The 

 coefficients of the powers of p are therefore constants. Inte- 

 grating to the time, 



jvcft=jz r</i+z '[r]+iz "[f]+ (21) 



If the current be steady at beginning and at end, 



j(V-z r)*=z„'[r], .... (22) 



and if the initial current be zero, and the final value be C, 



j"(V-Z I>< = Z 'C; (23) 



so that Z/C is the electromotive impulse employed in setting 

 up the magnetic and the electric energy of the steady state 

 due to steady V at the terminals. Thus 



L =Z ' (24) 



finds the impulsive inductance from the resistance-operator. 

 Or, 



L =(Z-Z )p- 1 withp = 0. . . . (25) 

 In a similar manner, we may show that 



S =Y ' = -Z - 2 Z ' (26) 



