ULTRA-HIGH-FREQUENCY VACUUM TUBES 641 



and where the 5's given by (16), (17), (18), etc., are to be inserted in 

 the o's given by (21), (22), (23), etc., before the integration can be 

 carried out. 



The formal solution of the problem has been reached with the 

 attainment of (34), (35), (36), etc. As soon as specific functions are 

 chosen for the current, £K + (p"'(t)^kmele, the initial acceleration, 

 a„ + «(/), and the initial velocity, m„ + jLt(0, the integrations can be 

 performed to give the potential difference between two planes located 

 respectively at x = and x = x. 



While the general relation between current and potential is non- 

 linear, the first order current <p"'{t)elkme is linearly related to the first 

 order potential difference (Wa — Wbjelkm, and the results for this case 

 can be expressed conveniently by using complex functions in the 

 manner usual with electrical engineers. This will be done in some of 

 the following applications. 



In the treatment of second and higher-order components, it is 

 convenient to select the current components to correspond with powers 

 of the potential rather than vice versa. For example, (Va — Vb)i is 

 taken to be the complete expression for the fluctuating component of 

 potential, thus causing {Va — Vb)2, (Va — Vbja, etc. to vanish. Equa- 

 tion (36) then yields an expression for the second-order current in 

 terms of the first-order current, and (37) does likewise for the third- 

 order current. 



The general equations are, however, applicable to any converging 

 method of selecting the components, and the proper one for any 

 particular case is to be determined by considerations of simplicity and 

 convenience. 



Part II — First-Order Solution 



This is the linear case, so that to each component of current, A sin co/, 

 say, there corresponds a potential component of the form B sin (w/ + v)- 

 It follows that a current of the form Ae^^ will produce a corresponding 

 potential difference Pe^', and that p may be taken to be a generalized 

 exponent having the value ioi when sinusoidal currents are considered. 

 In the latter case, P will usually be complex. The generalized ex- 

 ponent p results in a more compact symbolism than would be possible 

 with the imaginary exponent, ico. 



Thus, we write for the current 



^^-K + W"(t) =K + Je^K (38) 



In a similar way the initial fluctuating components of acceleration and 



