24. Prof. Maxwell on the Motions und Collisions 
of the entire system of particles which must be compounded with 
the motion of the particles relatively to one another. We may 
call the one the motion of translation, and the other the motion 
of agitation. 
Prop. V. Two systems of particles move each according to the 
law stated in Prop. IV. ; to find the number of pairs of particles, 
re of each system, whose relative velocity lies between given 
mits. 
Let there be N particles of the first system, and N’ of the 
second, then NN! is the whole number of such pairs. Let us 
consider the velocities in the direction of x only; then by 
Prop. IV. the number of the first kind, whose velocities are be- 
tween z and x+dz, is 
N : =e #dz, 
aw or 
The number of the second kind, whose velocity is between x+y 
and #+y+dy, 1s 
if _ (ety? 1 
= —e€E B? 
B WV 1 Ys 
where £ is the value of « for the second system. 
The number of pairs which fulfil both conditions is 
I 
a2 (wt+y)? 
~ Kaa Dieta 
Pome es 
NN oan” 
Now 2 may have any value from —o to +o consistently with 
the difference of velocities being between y and y+dy; therefore 
integrating between these limits, we find 
; 1 pled d (5 
—————<—<—<—_——= € a2+ B2 . . . . 
sat V 02 + B? or = ) 
for the whole number of pairs whose difference of velocity lies 
between y and y+dy. 
This expression, which is of the same form with (1) if we put 
NN! for N, «?+? for a, and y for 2, shows that the distribu- 
tion of relative velocities is regulated by the same law as that of 
the velocities themselves, and that the mean relative velocity is 
the square root of the sum of the squares of the mean velocities 
of the two systems. 
Since the direction of motion of every particle in one of the 
systems may be reversed without changing the distribution of 
velocities, it follows that the velocities compounded of the velo- 
cities of two particles, one in each system, are distributed accord- 
ing to the same formula (5) as the relative velocities. 
Prop. VI. Two systems of particles move in the same vessel ; 
