of Perfectly Elastic Spheres. 27 
between 7 and 7+dr is 
1 r (r—v)? ere? 
N =s(e€ # —e «@ )dr=n, 
and the number of pairs which approach within distance s in 
unit of time is 
nn'rrs?, 
(v—r)2 (v+r)2 
v2 
=NN', str?ve B{e- aa a |r do. 
By the last proposition we are able to integrate with respect 
to v, and get 
be; it 
NN’ SA ee s*r8e° 02+ @ dr. 
(a? + B*)* 
Integrating this again from r=0 to r=0, 
QNN Va Vertes... (1 
is the number of collisions in unit of time which take place in 
unit of volume between particles of different kinds, s being the 
distance of centres at collision. The number of collisions be- 
tween two particles of the first kind, s, being the striking 
distance, is es VE 
2N?2 Var V 2a?s,?; 
and for the second system it is 
2N!? or V 28? 5,2. 
eur 2 
The mean velocities in the two systems are —= caer pet eS 3 so 
5 T 7 
that if /; and /, be the mean distances travelled by particles of 
the first and second systems between each collision, then 
1 te 24 92 
7, =7N, V2s,2+7N, ve Pe 
1] OUD ee = 
r=, vente +N, V25,%. 
2 
Prop. X. To find the probability of a particle reaching a 
given distance before striking any other. 
Let us suppose that the probability of a particle being stopped 
while passing through a distance dz, is adv; that is, if N_par- 
ticles arrived at a distance x, Nadwv of them would be stopped 
before getting to a distance w7+dz. Putting this mathematically, 
or (N= Ce-—, 
dx 
