210 Mr. E.V. Appleton and Dr. B. van der Pol on 



through the element dx o£ the filament and let the emission; 

 from this element be (S i a dx. 



Fis-. 8. 



x*o x=2 



Then (see fig. 



A'je-O— 1. 



When the anode circuit is closed and the emission collected 

 the temperature of the element dx will change from T to 

 T + ST, the resistance from r to r Q -\-Br and the filament 

 current (apart from the superposed emission current) from 

 if to i/o + Bif, where BT and Br are functions of x. 



If c is the thermal capacity per unit length of filament, 

 we have as the expression of the energy equilibrium of the 

 element dx in the new state * 



c^(8T) + <T + 8T)H^^i«=(/ fo + ^+^W 2 Oo + Sr), (3) 



which after neglecting squares of small quantities and 

 substituting from (2) becomes 



c~ (BT) + 4sT 3 ST + /3</u' a = / /o 2 Br + 2i fo Bi f . r + 2i fo i a kr . 



. . . ay 



Now the terms ST, Br, and Bif are related as follows : 



Br=a r BT, 



where a is a constant, and, if no series resistance is included. 

 in the filament circuit, 



1 Br . dx 



I r . dx 



Jo 

 * Neglecting the small correction due to the Kelvin effect. 



