642 BELL SYSTEM TECHNICAL JOURNAL 



velocity are taken to be 



(39) 

 (40) 



With this nomenclature, all terms appearing in the first-order 

 equations will contain the factor e^', which may accordingly be omitted 

 throughout. Thus, instead of writing Ge"' for the acceleration at the 

 " a " plane, the single symbol on will be used, where the omission of the 

 functional notation ai{t — T) indicates that the acceleration is taken at 

 the " a " plane where T is zero, and that the multiplying factor e"' is 

 understood. In a similar way the symbol /xi will indicate the first- 

 order fluctuating component of velocity at the " a " plane with the 

 factor gp' understood. 



A still further symbolism will be of assistance. The quantity pT 

 will be denoted by jS. When p is the imaginary ico then wT is the 

 transit angle, 9, which has been defined in previous papers,^- ^ and in 

 the sinusoidal case jS = id. 



With this nomenclature, the first-order potential difference, (35), 

 acceleration, (22), and velocity (28) may be written: 



(Wa - TOl 



+ paa 



^' 



J_ 



+ 



K 



6 



- -":; [fl„(/3e-^ + e- 

 P 



/? - 2e-P - |3g-^ + 2 

 + g-^ - 1 ) + phiai^ + e-^ 

 ^ - 1) -f ii^pie-^ - 1)] 



ai 



e-p - 



K 



p'^Uo 



^?b- 



Ul = —9 



^e- 



1 - e- 



_ ^ 

 puo 



+ p LK(^e- 



^e-& -^e- 



+ e-^ - 1)], (41) 



Ilia 



e-" - 



KT 



IJ.\ e' 



Mo 



+ 



1 



+ Ml 



2^ ^^' 



1 



2 ^ 



)] 



(42) 



(43) 



An alternate method of procedure, which is particularly useful at 

 moderately low frequencies where the transit angle d is only a few 

 radians, is to expand functions of (/ — T) into power series in T. The 



