MR. CLERK MAXWELL OX THE DYNAMICAL THEORY OE GASES. 
57 
internal vibration, or any form of energy other than simple motion of translation, these 
results will be modified. The value of 6 and the final velocities of the molecules will 
depend on the amount of internal energy in each molecule before the encounter, and 
on the particular form of that energy at every instant during the mutual action. We 
have no means of determining such intricate actions in the present state of our know- 
ledge of molecules, so that we must content ourselves with the assumption that the value 
of 0 is, on an average, the same as for pure centres of force, and that the final velocities 
differ from the initial velocities only by quantities which may in each collision be 
neglected, although in a great many encounters the energy of translation and the internal 
energy of the molecules arrive, by repeated small exchanges, at a final ratio, which we 
shall suppose to be that of 1 to (3 — 1. 
We may now determine the final velocity of after it has passed beyond the sphere 
of mutual action between itself and M 2 . 
Let V be the velocity of Mi relative to M 2 , then the components of V are 
h — li, Vi — flay — 
The plane of the orbit is that containing V and b. Let this plane be inclined <p to a 
plane containing V and parallel to the axis of x ; then, since the direction of V is turned 
round an angle 23 in the plane of the orbit, while its magnitude remains the same, we 
may find the value of ^ after the encounter. Calling it 
li) 2 sin 2 3-{-\/ (%— fli)*+(?a— (TO* s in 2d cos <p). . . (1) 
There will be similar expressions for the components of the final velocity of Mj in the 
other coordinate directions. 
If we know the initial positions and velocities of M! and M 2 we can determine V, the 
velocity of relative to M 2 ; b the shortest distance between M, and M 2 if they had 
continued to move with uniform velocity in straight lines ; and <p the angle which deter- 
mines the plane in which V and b lie. From V and b we can determine 6, if we know 
the law of force, so that the problem is solved in the case of two molecules. 
When we pass from this case to that of two systems of moving molecules, we shall 
suppose that the time during which a molecule is beyond the action of other molecules 
is so great compared with the time during which it is deflected by that action, that we 
may neglect both the time and the distance described by the molecules during the 
encounter, as compared with the time and the distance described while the molecules 
are free from disturbing force. We may also neglect those cases in which three or more 
molecules are within each other’s spheres of action at the same instant. 
On the Mutual Action of Two Systems of Moving Molecules. 
Let the number of molecules of the first kind in unit of volume be Nj , the mass of each 
being M,. The velocities of these molecules will in general be different both in magni- 
tude and direction. Let us select those molecules the components of whose velocities 
MDCCCLXVII. i 
