26 Prof. Maxwell on the Motions and Collisions 
find the number of these which it approaches within a distance 
s in unit of time. 
If we describe a tubular surface of which the axis is the path 
of the particle, and the radius the distance s, the content of this 
surface generated in unit of time will be zrs*, and the number 
of particles included in it will be 
Nave aes) 
which is the number of particles to which the moving particle 
approaches within a distance s. 
Prop. VIII. A particle moves with velocity v in a system 
moving according to the law of Prop. IV.; to find the number of 
particles which have a velocity relative to the moving particle 
between r and 7+ dr. 
Let w be the actual velocity of a particle of the system, v that 
of the original particle, and 7 their relative velocity, and @ the 
angle between v and 7, then 
u2=v? + r2— 2vr cos 8. 
If we suppose, as in Prop. IV., all the particles to start from the 
origin at once, then after unit of time the “ density ” or number 
of particles to unit of volume at distance w will be 
From this we have to deduce the number of particles in a shell 
whose centre is at distance v, radius = r, and thickness =dr, 
Ly oe EE 
N "te oe —e a har, baat (~))) 
avr 
which is the number required. 
Cor. It is evident that if we integrate this expression from 
r=0 to r=, we ought to get the whole number of particles 
=N, whence the following mathematical result, 
© (ena (eta 
\ dz.a(e @ —e” @ \= Vara, ... 410) 
0 
Prop. IX. Two sets of particles move as in Prop. V.; to find 
the number of pairs which approach within a distance s in unit 
of time. 
The number of the second kind which have a velocity between 
v and v+dbv is 
4 ee 
N! a Va ve Bdv=n. 
3 / or 
The number of the first kind whose velocity relative to these is 
