the Form of Free Triode Vibrations. 211 



We may assume as a rough approximation that the 

 original resistance r is equally distributed along the fila- 

 ment. Thus by substitution lor hr and Si/- from (4) we 

 have 



C/^?+45 f T 3 . ST . dx-Y </>i a = -i f0 2 a E £+2i fo i a B*k, 



' . . . (5) 



where 



Cf = lc, the thermal capacity of the whole filament, 

 R = /r , the original resistance of the whole filament, 



"?£**■ 



the mean distribution of emission 

 current through the filament, 



and f = f-\ ST.dx, the mean temperature variation of 

 Jo the filament. 



Putting further, 



s f T 3 . ST . dx =y{ T 3 . ST . dx = S T' 3 f, . (6) 

 Jo * Jo 



where T' is a kind of mean temperature the exact value of 

 which obviously cannot be stated, and substituting for s 

 from (2), we may write (5) in the form 



G/f + fyR«(^+«)?=n.(2*E r ^), . . (7) 



where E/ is the filament potential difference and T 4 is the 

 mean of the fourth power of the temperature when no anode 

 current is allowed to flow. 



Obviously there are some important limitations to the 

 use of the formula (7) arising from our lack of knowledge 

 regarding the temperature and emission current distribu- 

 tion over the length of the filament. However, some 

 general statements can be made, such that k must lie be- 

 tween ±1 and that, when no series resistance is included 



in the filament circuit, the value of (-= + rt ) * s positive. 



Without a series resistance the emission current on entering 

 the filament circuit is probably divided more or less 

 equally between the two paths to the filament, in conse- 

 quence of which the value of k will be small. Hence the 

 value of (2~kE f — <j>) is negative unless a large filament P.D, 



