640 BELL SYSTEM TECHNICAL JOURNAL 



+ T^,"'{t - T)b, - \ T<p,""{t - TW + Ta,(t - T) 



- Ta.'it - T)b2 - Ta^'it - T)b, + \ Ta,"{t - T)8r'' 



+ ai(/ - T)82 + a2(t - T)8, - a,'{t - T)8,- + fx^^t - T) 



- Mi'(/ - r)52 - M2'(/ - T)8, + ^Mi"(^ - T)b,\ (30) 



Aside from their length these equations are not compHcated and in 

 appHcations to special cases many terms are apt to vanish, leaving 

 relatively compact expressions. 



In circuit work the potential difference between the two parallel 

 planes, " a " and " &," say, is more often required than the electron 

 acceleration. This may be found from the definition of the potential 

 difference, namely 



Va- Vi,= C Edx, (31) 



in which / remains constant during the integration. From (10) 



dx = I K — + aaT -\- Ua j dT = UodT, (32) 



so that with the aid of this equation and (2) the potential difference is 

 given by: 



Wa- Wb^ r adx = r a ( K ~ -\- aaT -{- Ua) dT, 



(33) 



where the symbol, W, is used as an abbreviation for eVjkm. 



In the same way as the acceleration and velocity are divided into 

 components, the potential difference may be split up into {Wa — Wb)o, 

 {Wa — Wb)i, (Wa — Wb)2, etc. which are defined by: 



(Wa - Wb)o = r aoUodT, (34) 



Jo 



(Wa - Wb)i = r UiUodT, (35) 



Jo 



(Wa - Wb)2 = r a2UodT, (36) 



Jo 



(Wa- Wbh= Ca^uodT, (37) 



Jo 



