VACUUM TUBE ELECTRONICS 



85 



When Ua and Ub are each separated into their components accord- 

 ing to (5), so that (44) may be resolved into a system of equations, 

 we have for the first two equations, analogous to (6) and (7), 



47r-/oS = (a - h) 

 m 



ox \ ax 



(45) 



and 



ox \ ox ox / di~ dx \ dx 



+ (a-b) 



dx \ dt / dt\ dx dx 



, (46) 



where the components of Ut have been expressed in terms of those of 

 Ua by means of (40) and the relation that Ubo = — Uao- 

 The solution of (45) is, as before, 



where 



Uo = a.r-'^ 



= IStT — /oa — 



\ m a 



(47) 



Before attempting to solve (46) we make a change of variable as in 

 (10), writing 



^^M^ 



a 



and 



U,= 



This gives from (46) 



P' d'-o: 1 3-0) 



^ ^ ^ ^ d^dt j 



(48) 



In finding a solution for this, we shall restrict ourselves to the case 

 where all of the electrons turn back at the virtual cathode, so that 

 a = b and therefore the last term of (48) vanishes. The solution of 

 the remaining equation is then, 



f/i 





sin pt + \ F,{i^ + pt) + 7 F^iii - pi) 



(49) 



which is analogous to (12). 



Again, assuming the two arbitrary functions to have the form, 



Fiii^ + pt) = a sin (z^ -}- pt) + b cos (/t + pt), 

 Pikik - pi) = c sin {i^ - pt) -\- d cos (/^ - pt), 



(50) 



