VACUUM TUBE ELECTRONICS 67 



In mathematical form the two boundary conditions may be set 

 forth as follows: 

 when, 



^ = 0, Ui must be finite, (14) 



^ == li, Ui = Msmpt + N cos pt. (15) 



From (12) and (13) these result in the values: 



c = 0, d = - 2, 



a = |- (M cos ^1 - A^ sin ^i) + cos ^1-71 sin ?i, (16) 



6 = I (1 - cos ^1) - sin ^1 - ^ (Msin ^1 + TV cos ^1). (17) 



Thence from (12) we have for the alternating-current velocity, in 

 general , 



Ui = {M + iN) (cos ^1 + i sin ^1) (cos | - i sin ^) 



+ ^2 {(cos ^1 - |sin lij - i (I - |cos ^1 - sin ?ij| 



(cos ^ - ? sin ^) - (1 - |sin A - i- (1 - cos ^)] , (18) 



where, in accord with engineering practice, complex notation is em- 

 ployed, so that sin pt has been replaced by e'^' and cos pt has been re- 

 placed by ie^p\ where i = V— 1. 



The first step in the derivation of fundamental relations has now 

 been achieved. The alternating-current velocity at any point between 

 the two planes has been expressed in terms of the alternating-current 

 velocity, M -f iN, existing at a definite value of x, say Xi, correspond- 

 ing to the transit angle ^1. 



The next step is a determination of the potentials corresponding 

 to the velocities Uo and Z7i, respectively. Thus from (1) and (2) 



_£^=^+y«' (19) 



m dx at ox 



and then with the separation of components as given by (5) 



