﻿596 Prof. 0. W. Richardson : Some Applications 

 and x is the least positive root o£ 



Thus a is a function o£ 6 and s only. 

 Evidently we may replace (1) by 



n (5*)^ r~£y(& % d t dr i d £ u da d f- • • ( 2 ) 



Now consider the reverse collisions which bring new 

 electrons into the group. Each orbit lies in a plane, the 

 plane containing b and r, and is symmetrical about a plane 

 perpendicular to that of the orbit containing the apsidal 

 distance. The magnitude o£ r is constant throughout a 

 collision, the effect being simply to rotate r through the 

 angle 26. The various components f, tj, f will, however, be 

 changed, let us say to f, i/, ?'. The new values may be 

 written down by making use o£ the fact that the component 

 of velocity along the apsidal distance has been reversed 

 whilst the perpendicular component is unaltered. Thus,, if 

 -^ = is the plane containing r and £, 



£' = I - 2% sin 2 6 + VV + V sin 26 cos f, . (2 a} 



with similar expressions for rj and £\ 



The particles /({■ , 77, f) <7£ e/?? df may be represented by a 

 distribution of points occupying an element of volume 

 di; d7]d£ in* a three-dimensional velocity diagram. The 

 volume d£' drj' ' d£' occupied by the deflected points will be 

 equal to dgdrjdg, since the new points may be obtained 

 by reflecting the undeflected points in a plane perpendicular 

 to the orbit and tangential to it at the apse. Thus, in 

 considering the reverse collisions which bring extraneous 

 electrons into the group /Tf ?? f) d%dr)d£, the only change we 

 require to make in (2) is the replacement of /(f, rj 1) by 



/tr.v.r). 



In this way we arrive at the equation 



... (3) 



\ or the stationary state in which /is independent of y and z.. 

 The left-hand side is the rate of change of / owing to 

 collisions, and the right-hand side that which arises from the 

 displacement of the group of electrons as a whole. Following. 



