ULTRA-HIGH-FREQUENCY VACUUM TUBES 



637 



and similarly functions of /„ may be written 

 Kta) =f{t-T-8)= f(t -T)- fit - T)5 



T)8'' 



(13) 



When (10), (11), (12) and (13) are used in conjunction with (8) the 

 result is a relation between t, T and 5 as follows: 



= ^[3r25 + 3r62+53] 



+ <p{i) 



-(r+5) 



^(t-T)-<p'it-T)8^~<p"(t-T)8'^- 



+ ^{T^-{-2T8+8'-) 



^"{t-T)-^"'{t-T)8^j^.p""{t-T)S- 



'.{t-T) 



+(r+5) 



■a'{t-T)8+j^a"{t-T)8-'~ 



1 



^{t-T)-,x'{t-T)8-Vj,,x"{t-T)8'- j 



(14) 



This equation may be written in the form of a power series in 5. It 

 cannot yet be solved directly for 8. It has the advantage, however, 

 that 5 is not involved in the transcendental functions (p, a and ix, so 

 that an indirect method may be used. Let 5, (p, a and fx each be split 

 up into series as follows: 



(15) 



5 = 5i + 52 + 53 + etc. 



<p = <P\ -{- <P2 + <Pi -{- etc. 



a = a\ -\- ai -\- otz -\- etc. 



II = /ii + ^i2 + M3 + etc. 



These are to be substituted into (14) and the resulting expression may 

 then be expressed as an infinite series of separate equations such that 

 the first equation includes all linear terms which have the subscript 1, 

 but no other terms; The second equation includes all linear terms 

 having the subscript 2 and also all quadratic terms having the subscript 

 1; the third equation includes all linear terms having the subscript 3, 

 cubic terms with subscript 1, and also products of quadratic terms with 

 subscript 1 and linear terms with subscript 2. The rules for succeeding 

 equations are analogous, so that in general, the sum of the subscripts 

 of each term of the wth equation is equal to n. 



