644 BELL SYSTEM TECHNICAL JOURNAL 



motion and reverse their direction. When this happens, (1) is no 

 longer applicable because the velocity at a given point has become 

 multi-valued. This restriction was pointed out by Miiller ^ as limiting 

 any analysis which starts with (1). 



It may be concluded then that the inherent limitation on (41), (42) 

 and (43) is a singly valued velocity rather than the behavior of the 5's 

 in the neighborhood of a point where the acceleration is zero. 



Part III — Second-Order Solution 



The second-order solution has practical utility in the computation of 

 distortion and of the modulating and detecting properties of ultra-high- 

 frequency thermionic systems. Even in low-frequency applications 

 such computations are long and tedious. The introduction of the 

 added complication of appreciable transit angles causes further diffi- 

 culty because of the unwieldy length of the equations. Accordingly, 

 instead of a general exposition of second-order effects, a greatly 

 simplified special case will be treated at the present time, leaving the 

 details of a more general solution until the need for it has become more 

 acute. 



The rectifying properties of a parallel plane diode operating with 

 complete space charge will be calculated. The complete space charge 

 condition is defined by placing initial velocities and accelerations 

 equal to zero. When this has been done in (16), (17) and (23) and the 

 resulting values of the 5's have been substituted in (23), functions of 

 (/ — 7") may be expanded into power series to give the following: 



_ ^ (- r)"+V2("+^>(0 



, 1 f f (- r)"+"'+Vi(»+^)(0y,,("'+3)(^) 



'^ K to to {n -f- 3) (m + 3)n\ml ' ^ ^ 



The second-order potential difference is given by (36) where «o 

 = KT^I2 for complete space charge. Thus, from (44) 



n=0 (W -H 4) ! 



_ 1 " " (- r)"+"'+Vi("+«>(/)y !("'+«)(/) . 



2 nto to {n + m + 4)(« -}- 3)(w -f 3)n\m\ ' ^ ^ 



The second-order potential difference (Wa — Wb)2 will now be taken 

 as zero, which implies that the first-order potential difference only is 



