636 BELL SYSTEM TECHNICAL JOURNAL 



A third integration gives 



where x is zero when t — ta. 



This choice of initial conditions allows one of the two parallel planes 

 to be located at the position where x is zero and where the electron 

 velocities and accelerations are given as above in terms of the time 

 instant ta when the electron crosses the plane, which may be referred to 

 as the " o " plane. When these initial conditions are specified, then 

 (6), (7), and (8) allow the acceleration, velocity and position, re- 

 spectively, to be determined at any time /, thereafter. 



These quantities are expressed in terms of t and ta whereas it is more 

 convenient in vacuum tube work to have the acceleration and velocity 

 expressed in terms of t and x. Ideally, this could be done by solving 

 (8) for ta and thence eliminating ta from (6) and (7). Practically, (8) 

 cannot be solved directly for ta because it is a higher order equation and 

 involves (p, a and /x which may be (and usually are) transcendental 

 functions of ta. However, an indirect method can be employed. 



If (p, a and M were zero, then x would be given by the relatively simple 

 equation 



^ ^ ^(LZ^ + aa^^^ + Ua{t - ta). (9) 



Although this is a cubic, nevertheless {t — ta) may be obtained with 

 relative ease in any particular case. The use of a new variable T to 

 replace x is suggested by (9) and accordingly the defining equation of T 

 under all conditions is taken to be : 



X = K— -{- aaY + ^<^^> (10) 



which holds even when <p, a and ju are not zero. It is evident when 

 (f, a and fx are small that T does not dififer very much from t — ta as 

 (10) must then become nearly equivalent to (9). It thus seems 

 expedient to write in general 



t - ta= T-{-8, (11) 



and note that 5 becomes very small when (p, a and /x are small. 



On the basis of (11), functions of {t — /„) may be expanded into 

 series in powers of 8 as follows : 



At - O = fiT + 5) = f{T) +f'{T)8 + ^/'(r)52 + . • .. (12) 



