ULTRA-HIGH-FREQUENCY VACUUM TUBES 639 



These formulas together with (11) and (15) allow the acceleration 

 (6) and velocity (7) to be written in terms of t and T, and hence effect- 

 ively in terms of t and x, for 7" is a function of x only, as given by (10). 

 The resulting equations for acceleration and velocity are conveniently 

 broken up into a series of equations in accord with the same rules 

 formulated for obtaining the 5's from (14). Thus, we write 



o = ao + fli + ^2 + as + etc., (20) 



where ao comprises those terms having zero or no subscript, and obtain 

 from (6) : 



a,= KT+aa, (21) 



a, = Kb, + <pi"{t) - <p,"{t -T)+ a,{t - T), (22) 



a2 = Kh + <P^"{t) - <p2"{t - T) + W"{t - T)8x 



+ a^it - T) - ay'it - T)8u (23) 



as = K8s + <p,'\t) - ^s"{t - r) + W"{t - T)82 



+ <p2'"(i - T)8, - \W"'{t - T)8{~ + az{t - T) 



- a2'{t - r)5i - a,'(t - T)82 + \ a,"{t - T)8,\ (24) 



a4=---. (25) 

 Treating the velocity in a similar way we write 



w = Mo + Wi + M2 + uz + etc. (26) 



and obtain from (7) 



Wo = K Y + ^"-^ + "«' ^^^ 



Wl = KT8, + flaSl + ^l{t) - cpy'it - T) 



- T^,"{t - r) + Ta,{t - r) + ^i,{t - r), (28) 



U2 = KT82 + y 5l2 + aa82 + <P2{t) - <P2{t - T) 



- T^2"{t - r) + T^^!"{t - T)8, + Ta2{t - T) 



+ a,{t - T)8, - Ta,\t - T)8x + fX2(t - T) - mi'(^ - T)8u (29) 



W3 = XrS. + X5i52 + aa8z + <^3'(0 - <Pi\t - ^) 



+ ^ <p,"\t - T)8{' - T^^'ii - r) + T^,"'{t - r)52 



