832 Prof. 0. W. Richardson on Thermionies. 



We get the first-order terms by putting n=0. To this 

 approximation 



»=4wef (l + 2keY)e- 2keY . 



o K ' 



To the third order in a/b we have to add the terms corre- 

 sponding to w = l, which give 



4 



-r, ne c T . 

 6 b 



(1 + 2keY + 2k ^e 2 Y 2 )e- 2keV 



and so on. 



The value of the current when Y = 0, i. e. when all the 

 cylinders are at the same potential, will be given by putting 

 V = in the doubly infinite series for i. This gives 



= 4:ne ca tan -1 (a/b) (15) 



In this case there are no forces in the field so that this result 

 should be the same as that given by the general formula 

 (p. 822) 



where the surfaces A and B are the portions of two con- 

 centric circular cylinders of radii c and b which are cut off 

 by planes, perpendicular to the common axis, distant a from 

 each other, and where c is small compared with b. We then 

 have 



dS = cd</> d,z , dfS = bdcj) dz, cos n*r = b/r, 



d(j> ^ =dz Q — r , 



Hence 



* = Snetfc f Fm 2 Tpe-*^ 2 fltyf dz\ „ 9 L /~~° .^ 

 Jo Jo • o \° +\ z — z o) ) 



= 4weac tan - 1 y- 

 o 



in agreement with the previous result. 



Formulae (14) and (15) are subject to the restrictions 



that clb and 7 * 7 are small, [f these conditions do not 

 o — b\ 



hold the series become much more cumbersome. 



