ELECTRON STREAMS IN A DIODE 839 



where A (5) is an arbitrary finite function of S, g is an arbitrary constant, 

 and the coefficients gi and g2 have such values that the power terms in / 

 cancel out in (15). The finite function may, for example, be a sinusoidal 

 function of 5, or a series of such sinusoidal terms. The values of the coefi&- 

 cients are easily calculated, and when the resultant expression forFi(5) is 

 substituted in (15), it may be written: 



U=, + A^S)+l[g.^+IJl.dt] (22) 



where 5 is 5 with the Id term omitted, that is, 



S = eE^ j I^dt (23) 



£n a similar manner it may be shown that, for x to be finite, (18) assumes the 

 form 



X = k + B{S) - 



'-^ *-?[£-///"'] « 



where k is an arbitrary constant, and B(S) is any arbitrary finite function of 

 S. Then (22) and (24) constitute the general solution when a continuous 

 direct current is flowing in the diode. They are mathematical means for 

 shortening the calculations in the presence of the direct current. 



It is believed that the general solution presented in this section will serve 

 as a guide for reasoning about electron streams, and as a guide that can be 

 used in particular problems. It should also be an aid for considering the 

 large signal theory of electron streams. But it is here advisable to confine 

 attention to a less ambitious program, and apply the method to the case 

 of small signals. The results will not be entirely new, but they will bring 

 out certain important features of the general solution. 



3. The Small Signal Theory oe Electron Streams 



The small signal theory is developed as follows : The value of each depend- 

 ent variable, in any plane normal to the rc-axis, is regarded as the sum of 

 two components: a value that does not vary with time and is therefore 

 called the d-c. component, and a value that does vary with time and is 

 called the a-c. component. All of these components may vary with x, that 

 is, with the exception of the total current density which is a function of 

 time alone. Corresponding to small signal circuit theory, it is also assumed 

 that the a-c. quantities are small compared to the d-c. quantities, and that 

 the squares and products of the a-c. quantities are negligible in comparison 

 to their first order values. For such signals the circuit equivalent of a diode 



