62 
ME. CLEEK MAXWELL ON THE DYNAMICAL THEOEY OF GASES. 
be integrated 
where Q is ■some function of ??, £, &c., already determined, and f 2 is the function 
which indicates the distribution of velocity among the molecules of the second kind. 
In the case in which n= 5, V disappears, and we may write the result of integration 
qn 2 , 
where Q is the mean value of Q for all the molecules of the second kind, and N 2 is the 
number of those molecules. 
If, however, n is not equal to 5, so that V does not disappear, we should require to 
know the form of the function f before we could proceed further with the integration. 
The only case in which I have determined the form of this function is that of one or 
more kinds of molecules which have by their continual encounters brought about a 
distribution of velocity such that the number of molecules whose velocity lies within 
given limits remains constant. In the Philosophical Magazine for January 1860, I have 
given an investigation of this case, founded on the assumption that the probability of a 
molecule having a velocity resolved parallel to x lying between given limits is not in any 
way affected by the knowledge that the molecule has a given velocity resolved parallel 
to y. As this assumption may appear precarious, I shall now determine the form of the 
function in a different manner. 
On the Final Distribution of Velocity among the Molecules of Two Systems acting on one 
another according to any Law of Force. 
From a given point O let lines be drawn representing in direction and 
magnitude the velocities of every molecule of either kind in unit of 
volume. The extremities of these lines will be distributed over space 
in such a way that if an element of volume dV be taken anywhere, the 
number of such lines which will terminate within cZVwill be f(r)dV , 
where r is the distance of dV from O. 
Let OA=a be the velocity of a molecule of the first kind, and OB = 5 that of a mole- 
cule of the second kind before they encounter one another, then BA will be the velocity 
of A relative to B ; and if we divide AB in G inversely as the masses of the molecules, 
and join OG, OG will be the velocity of the centre of gravity of the two molecules. 
Now let OA'=«' and OB'=Z»' be the velocities of the two molecules after the 
encounter, GA=GA' and GB=GB', and A'GB' is a straight line not necessarily in the 
plane of OAB. Also AGA'=20 is the angle through which the relative velocity is 
turned in the encounter in question. The relative motion of the molecules is com- 
pletely defined if we know BA the relative velocity before the encounter, 20 the angle 
through which BA is turned during the encounter, and <p the angle which defines the 
direction of the plane in which BA and B'A' lie. All encounters in which the magni- 
tude and direction of BA, and also 0 and <p, lie within certain almost contiguous limits, 
