On Thermionics. 



821 



Then the number whose velocity components lie between 

 z and z-\-dz is 



nf(z) dz = ni^~je- kmz ' 2 dz. 



The numbers which have components between <p and 

 (jy + dcf), and for which at the same time 6 lies between 6 and 

 + dd is 



n<j> cos 6 F(c/> cos d) /(<£ sin 0)<f>d<j> d6 



2n( — V ^ e"^ 2 cos dj> dd. 



The number for which <f> lies between <£ and <j> + d<p and 

 for which has any value will therefore be 



n<j> 2 d<j> F (£ cos 6) f(<j> sin 0) cos 6 dd 



J-v/2 



We shall now consider the application of the considerations 

 which have been brought forward to a number of particular 

 problems. 



§ 5. No Electric Forces. 



When there are no forces in the field the general problem 



becomes comparatively simple, as the trajectories are straight 



lines, radiating from the point of origin, whose direction 



cosines are proportional to the initial velocity components. 



Let dS (fig. 1) represent an element of the surface A, 

 ■«o being the normal to the element. Let dS be the origin 

 of polar coordinates of which n is the polar axis. Let the 

 element d$ 1 be cut off from B by the planes <£ and (p + d<f) 

 containing n and the cones whose semi-angles are and 

 Phil. Mag. S. 6. Vol. 17. No. 102. June 1909. 3 K 



