440 DEIJ. SYSrilM TECHNIC.1L JOURNAL 



wliere the bar over a symbol denotes the conjugate imaginary of the 

 same symbol unbarred. If this current flows through a circuit con- 

 taining resistance, self-inductance, and capacity in series, we have 



e = i?/+L^ + ^ fidL (23) 



Substituting for I its equivalent, as given by (21) or (22), we may 

 write the result in abbre\iated form as follows: 



e = (Shiin + Zhiu,) + (zkhk + Zkiik) H (24) 



where 



Cjn 

 Zn^R-Ljn-^- 



When the current flows through a network of impedances, we may 

 always write the equivalent series impedance of the network. Hence 

 equation (24) may be extended to cover the general case. It will be 

 noted that lower case 2's have been used to represent impedances in 

 the above discussion. Throughout this paper the attempt has been 

 made to employ the lower case letters to denote quantities which 

 involve time, reserving the capitals for those which do not involve 

 time. With this understanding, Z denotes a resistance, while z 

 represents a general impedance, which, of course, varies with the time 

 variation of a voltage impressed on it. With the aid of (24) we are 

 in a position to treat non-linear equations by the complex method. 

 Thus, omitting conjugates e- becomes, 



e~ — 2;,'?2(2/!) -\-Zk-l2i2k) + 2S/jS/,i2(0/,) + -Zh':kl2[h+k) 



-\-2zhZki2{h-k)-\-'2zkZki2{0k)-\- 



(25) 



which ma\' be written 



e- = e2(2h)-^e2(2k)-'re2(0h)-\-e2(h+k)-\-('2{h-k)-\-e2(0k)+ (2()) 



In (25) and (26) the significance of the double subscript notation is 

 brought out. The first syml)ol in the subscript refers to the order 

 of the term, and the second refers to the frequency. 



In the light of the foregoing discussion, the problem of writing the 

 general equations for the thermionic vacuum tube may be attacked. 



