On Thermionics. 



829 



a plane perpendicular to this axis. Let <p make an angle 

 with the normal OP, then PQ (fig. 3) will be the resolved 



Ffr. 3. 



part of the straight part of the trajectory on the plane per- 

 pendicular to z. It is clear from fig. 2 that ions will only 

 escape through the gap in the cylinder of radius b provided 

 they start from A within the limits comprised by the points 

 p and q. A comparison of figs. 2 and 3 shows that the 

 conditions for escape are 



(1) when P lies between c = and z=a, z/<j> must lie 

 between — r/PQ and (a — ~)/PQ, 



b-b, 



(2) when P lies between : = and c = 



z/<j> must lie between — z/FQ l and (a — :)/PQ. 



(3) when P lies between ,z = a and z = a( 1 4- y 1 — j- J, 



V b — bj 



i/</> must lie between — c PQ and (a—c)/PQ 1 . 



Qx Q and R are the points in which PQ intersects the 

 cylinders of radii b x b and d respectively. So that 



PQ = y/P—c 2 sm*e—c cos 6. 



PQ t = vV-f 2 sin 2 0-c cos 0. 



The region on the surface of A between : = and z = a may 



be referred to as the umbra ; the wings between ; = —a l ~ L 



and z=0, and between z = a and z — a ( 1 + * , J respectively 

 as the penumbrse. o Oi 



