23 Prof. Maxwell on the Motions and Collisions 
plane AGN, making NG a=N GA, andGa=G Aand Gb=GB; 
then by Prop. I. Ga and Gé will be the velocities relative to G ; 
and compounding these with OG, we have Oa and O23 for the 
true velocities after impact. 
By Prop. II. all directions of the line aGé are equally pro- 
bable. It appears therefore that the velocity after impact is 
compounded of the velocity of the centre of gravity, and of a 
velocity equal to the velocity of the sphere relative to the centre of 
gravity, which may with equal probability be in any direction 
whatever. 
If a great many equal spherical particles were in motion m 
a perfectly elastic vessel, collisions would take place among the 
particles, and their velocities would be altered at every collision ; 
so that after a certain time the vis viva will be divided among the 
particles according to some regular law, the average number of 
particles whose velocity lies between certain limits being ascer- 
tainable, though the velocity of each particle changes at every 
collision. 
Prop. IV. To find the average number of particles whose velo- 
cities lie between given limits, after a great number of collisions 
among a great number of equal particles. 
Let N be the whole number of particles. Let x, y, z be the 
components of the velocity of each particle in three rectangular 
directions, and let the number of particles for which a lies be- 
tween x and 2+dz be Nf(x)dz, where f(z) is a function of 2 to 
be determined. 
The number of particles for which y lies between y and y + dy 
will be Nf(y)dy ; and the number for which z lies between z and 
z+dz will be Nf(z)dz, where f always stands for the same 
function. 
Now the existence of the velocity z does not in any way affect 
that of the velocities y or z, since these are all at right angles to 
each other and independent, so that the number of particles 
whose velocity lies between 2 and zw + dz, and also between y and 
y + dy, and also between z and z+ dz, is 
Nfle) fly) fle)de dy dz. 
If we suppose the N particles to start from the origin at the 
same instant, then this will be the number in the element of 
volume (dx dy dz) after unit of time, and the number referred to 
unit of volume will be 
Nf(2) fly) fl2)- 
But the directions of the coordinates are perfectly arbitrary, and 
therefore this number must depend on the distance from the 
origin alone, that is 
Ae) fY) fl) = Pa" + 7? + 2"). 
