816 Prof. 0. W. Richardson : 



It is to be borne in mind that the equations for x x y x z r 

 will not in general be of the first degree, so that there will 

 be a number of roots corresponding to the successive real 

 and imaginary intersections of the surfaces -x/r, cf> l and </> 2 . 

 In any case the path of the particle will end as soon as it 

 has reached the conducting surface B, and if this surface 

 includes the whole of the analytical surface ty(x y z) — the 

 root to be chosen is that real root which corresponds to the 

 shortest time of transit from w y z . The proper root can 

 usually be easily picked out in simple cases. If the surface 

 B is only a part of the analytical surface -^ = bounded by 

 a curve or curves, it may in general be necessary to include 

 roots corresponding to any number, less than that of the 

 degree of the equations, of previous intersections of the tra- 

 jectory and the surface ^ = 0. The problem is then much 

 more complicated. 



The equations (3) and f4) may be solved for v and v 



giving 











u = ^{xyz 



*o 3A>^o wo), • • 



• • (5) 





Vq = 4>*( x yz 



x y z w ). . . 



• • (6) 



The equation c£ 3 = constant together with yfr(xyz)=z0 will 

 determine a curve lying in the surface i/r which contains 

 the points of intersection of all trajectories for which u and 

 w are constant. Similarly 4 = constant determines a curve 

 corresponding to constant values of v and iv . If f and rj 

 denote lengths laid out along- the normals to the level surfaces 

 of u and v at any point, then 



> ■ ■ ■ O) 



~b% V \ d,e / \dy ) \ ~dz ) 



WV{Tx) + Vdy) + V^) J 



The number of particles which are emitted in unit time 

 with velocity components between u and u + dn and v and 

 v -t-dv respectively may be denoted by /i(w )<i?/ and 

 f 2 {v () )dv where f\ f 2 are functions which will be discussed 

 later. For a constant value of w the number which simul- 

 taneously have velocities between the above ranges will be 

 proportional to/i(w ) ./^o) du dv , and these will fall on an 

 area dS of the surface ^ = 0, where 



7Q / a o\ du dv 1 

 ab cos («o l 'o n ^>) — s; — ^ — x ~ a~ 



where u v A n$ is the angle between the normal to the 



