Heat produced by Emission of Ions from Hot Bodies, 631 



flow o£ electrons, which constitutes the thermionic current 

 Avhen it is outside the filament, will distribute itself about 

 the Wheatstone's bridge circuit in a manner which depends 

 on the value o£ the various resistances and the position o£ 

 the return wire PQ. The actual current flowing at any 

 point of the filament will be due to the superposition of this 

 current upon that supplied by the battery B^ However the 

 additional current may distribute itself, there will he one 

 point in the filament where it contributes nothing to the 

 current i which flows along the latter. Let the value of i 

 at this point be i . Since the relation between currents 

 and potentials in a network is a linear one, i would be equal 

 to the value C of i when the thermionic current is prevented 

 from flowing, were it not for the change BRi in the re- 

 sistance R of the filament which is produced by the ther- 

 mionic current. The potential of the battery B x being V, 



we have C = Y/R and i = V/(R + BRi) = C (l- ^) 



to the first order in dRjRo- If p is the resistance per unit 

 length of the filament when the thermionic current is not 

 flowing, R= ^ p Q dx. If p ' is the value of p Q when the 

 thermionic current is escaping, R + dBi = J po'dx. As the 

 change in the resistance may be supposed, without serious 

 error, to be distributed uniformly along the filament, we 



may put p o '=p i 1+ -9=^1 V Taking the point where the 



current along the filament, when the thermionic current is 

 allowed to escape, is i as origin let the coordinates of the 

 ends of the filament be + x x and — x . The current which 



is heating the filament at any point x is then i + 1 jdx, j being 



the thermionic current flowing into unit length of the fila- 

 ment at any point. Thus the total rate of supply of energy 

 due to the Joule heating effect of the various currents when 

 the thermionic current is escaping is 



dx 



^J — X qJ x 



%.c;( 1 + ^)(i-f)[J>]'" 



+ r>( i+ T)[fH'"«- 



+ 2( 



