﻿606 Prof. 0. TV. Richardson : Some Applications 



A relation of type (26) holds not only for the complete 

 aggregate but for each individual range of velocity or 

 momentum used in specifying it. Thus if e(u.v .w)du dv dw 

 represents the number of particles emitted in unit time with 

 velocity components between u and u + du, v and v + dv, 

 w and w + dw, r(u . v . w) du dv dw the number of similar 

 particles in the reflected group and N(w . v . w) du dv dw the 

 •corresponding thing for the incident aggregate ; then 



€ {it . v . iv) = N(w . v . w) —r(u .v .w). . . (26 a) 



If this equation were not true, the character of the 

 statistically steady aggregate would change with time. 

 Now 



N(w . v . w) — N xf(u . v . w), 



where /is a probability expression calculable from Maxwell's 

 law. 



Both theory and experiment point to the conclusion that 



e (u . v . w) = e . N ,f(u . v . w) ; 



whence it follows that 



r (u . v . w) = v . N . f(u . v . to) . 



Thus the group which arises by the reflexion of a disordered 

 aggregate is itself disordered, or the process of reflexion 

 is of such a nature as to transform the velocity distribution 

 of the incident aggregate into the same distribution among 

 the reflected group. 



This relation may be expressed somewhat differently as 

 follows : — If n is the number of electrons per unit volume 

 of the surrounding space in the state of equilibrium, then 

 the number striking unit area of surface in unit time with 

 velocity components between u and u + dn, v and v-\-dv, 

 w and w-\-dw is 



3/2 



ite- km ^ +v2 +" 2 )dudvdw 



•ffi 



if u is normal to the surface. Out of this group let the 

 proportion reflected with velocity components between u x and 

 Ui + dui be F(u . v . id . wj) du^ the proportion with velocity 

 components between i\ and v-j-di\ be Gr(« . v w . v ± ) dv h and 

 the proportion with velocity components between w 1 and 

 ic Y + dw 1 be H.(u . v . w . w x ) dw lm Then in the steady state 

 at the temperature corresponding to k the total number of 

 electrons reflected from unit area in unit time with velocity 



