and Conductance Operators. 483 



To obtain the relations between B/ and K7, and 1/ and S', 

 we have 



Y=(R / + L'p)- 1 = (R'-L»I- 2 , .... (7) 



Z=(K / + S / p)- 1 = (K / -S»J- 2 i .... (8) 



from which we derive 



-I 2 S'=L', -J 2 L'=S', V . (9) 



J 



L'/R' = - S'/K'. R'/K' = I 2 = - I//S', . 



all of which are useful relations. 



4. By (3) and (4) we have the equations of activity 



VC = R'C 2 +p(p/C 2 ), (10) 



VC = K'V 2 + p(iS'V 2 ), (11) 



in general. Now, if we take the mean values, the differen- 

 tiated terms go out, leaving 



VC = R'C 2 =K'V 3 , (12) 



the bars denoting mean values. The three expressions in 

 (12) each represent the mean dissipativity, or heat per 

 second. R/ and K' are therefore necessarily positive. It 

 should be noted that R/0 2 or R' V 2 do not represent the dis- 

 sipativity at any moment. The dissipativity fluctuates, of 

 course, because the square of the current fluctuates ; but 

 besides that, there is usually a fluctuation in the resistance, 

 because the distribution of current varies, and it is only by 

 taking mean values that we can have a definite resistance at 

 a given frequency. 



If the combination be electromagnetic, and T denote the 

 magnetic energy, its mean value is given by 



T=iL'C 2 , (13) 



so that L' is necessarily positive and S' negative. But JL/C 2 

 is not usually the magnetic energy at any moment. 



If the combination be electrostatic, and U denote the elec- 

 tric energy, its mean value is 



U = iS'V 2 , (14) 



so that S' is positive and U negative. The electric energy at 

 any moment is not usually ^S'V 2 . 



But, in the general case of both energies being stored, we 

 have 



T_U = iL'(7=-iS / V 5 (15) 



2K2 



