On Thermionics. 831 



§ 9. Case of a Thin Hot Wire. 



Under certain restrictions the preceding expressions sim- 

 plify very considerably. In the first place we notice that 



if y — j- is small compared with unity the penumbrae may 



be left oat of account. This leaves us with only the central 

 one of the three integrals in square brackets and corresponds 

 to the case when the inner surface of the cylinder B is very 

 close to the outer surface of the hot cylinder A. Further- 

 more if c is small compared with b the projections PQ 

 (fig. 3) of the trajectories on a plane perpendicular to the 

 axis all become approximately radial, and we have, to this 

 approximation, PQ = b. Under these circumstances the 

 current to the outer cylinder will therefore be equal to 



it a—z ' 



i = hie c f cos 6 cW f Pro 1 4> 2 e~ km * 2 dj> f dz f e~ k ™ 2 dz. 



If alb is reasonably small only small values of z will be 

 important, since large values of <p occur only rarely and 

 much more rarely therefore will large values of z occur. It 

 is therefore appropriate to expand the exponential in powers 

 of 'z, giving 



a—z. 



» "■ „b $ 



i = 8nec\ k 2 m 2 ^ 2 e- km <t> 2 dcj> \ dz\ £(-l) n — — ^di 



V 3 i T »♦ 



= inec\ 2kh 



. « *< 1 m"a 2 <» +1) <i 2 »+ 1 



X 



By continued integration by parts we see that 



3 s=n-l 1 



07M n) 2)1-1 —kmQ 2 j J i -|/7 2\' !_1 --' 1 '"" 1 " V — r 



U m <f> e d<f>=\n-l(kma-) e ,^ (km a 1 ) 8 (n-l- s) 



Hence 



r m=cd / 1 W fl 2(n+l) rs=n+l 1 -i "j 



The number of integral values of n which it will be necessary 

 to take in a practical case depends on the smallness of a/b. 



