500 
MR. O W. RICHARDSON ON THE ELECTRICAL CONDTJCTTYITY 
of course, only about one-hundredth that of an air molecule possessing the same free 
path, owing to the great velocity of agitation (d‘the corpuscles. Nevertheless we are 
(|uite justified in assuming that the time during which the corpuscles are moving 
freely is great compared with that during which they are colliding. In fact this 
assumption follows at once if we are to attach any definite meaning to the ideas of 
free paths and collisions. 
If, in addition to neglecting the number of corpuscles which are colliding at any 
moment in compaiison with those which are not, we assume that the atoms of the 
metal and the positive ions oscillate about fixed centres and are subject to forces of 
restitution which are functions of their displacements only, we oldain at once, by the 
a})plication of the (ordinary analysis of the kinetic theory, the distri])ution of vel()city 
among the corpuscles. This is found to Ije the same as that for an equal number of 
similarly constituted gas molecules. Thus the number of corpuscles (N„Ni,.N„.) which 
have velocity components in three jnutually perpendicular directions between u and 
u -f du, V and v -f dv, and lo and w + dw res})ectively are given by 
and 
N„ = 
. = N 
/ knv 
1 
) du 
\ 77 , 
/ 
11 
f km 
i 
V 77 , 
. = N ( 
drnv 
f dw 
\ 77 ; 
N,„ = 
where N is the total number of corpuscles considered, m is the mass of a corpuscle, 
a, /3, and y are the impressed velocity components of the corpuscles in the direction 
(^f V, and w respectively, whilst is the average energy of translation of a 
corpuscle, and is equal t(j that of a gas molecule at tlie same temperature as the 
metal considered. The velocities a, y are connected with the components q, r, s of 
the current density according to the relation 
(y, r, s) = n (a, /3, y) e, 
where e is the charge on a corpuscle, and n is tlie number per cub. centim. 
§ 2. If we su])pose the impressed velocities to l)e nil or to be negligilde compared 
with the velocities of agitation, the number of molecules in unit vohnne having 
velocity conqxments between u and n (h(, v and i'dc, and iv and ivdtt' 
bec< >mes 
dr dir . (2), 
\ 77 / 
whilst the number with these velocity components which bit unit area })erpendiculai' 
to u per seco]id is 
U ( ^ (-'2+c+(<■■■') 
V 77 / 
(3). 
