FREE AND PERFECTLY ELASTIC MOLECULES IN A STATE OF MOTION. 
9 
In the first of these let f3 0 = ft, S = v, B = nT> ; then shall 
2 (v + 0) 
£=-/3 + 
n + 1 
or ft = - * 
n 
which evidently expresses the upward velocity given to the plane by the impulse of 
one molecule when the velocity of incidence and reflexion is the same. The plane 
ascends and descends the height due to this velocity, and then encounters the next in the 
succession of molecular impacts without any transference of force taking place between 
them; and n being taken an indefinitely great number, (3 is infinitesimal in respect 
to v, and the height through which the plane traverses is also infinitesimal, so that it 
is supported as if by a continuous force of upward pressure. The time between each 
impact is, according to the law of falling bodies, equal to the time taken by the force 
of gravity to destroy and reproduce the infinitesimal velocity v/n. This is 2 vjgn : the 
velocity which a free body gains or loses in a unit of time by the force of gravity 
being represented by g. The number of impacts in a unit of time is therefore 
gnj2v = A. This, then, is the relation between the weight of the plane, in terms of 
that of the molecule unity, and the rapidity of the succession of impacts necessary to 
support it in a condition of statical equilibrium. Now, if the plane forms part of 
the surface that encloses the medium and that counterbalances by its weight the 
effect of the impacts of the confined molecules, such effect must correspond with the 
succession represented by A; and we deduce that the elastic force of a medium, as 
represented by the weight or pressure required to confine it, is directly 'proportional to 
the number of molecular impacts that take place against a unit surface in a unit time 
with a constant velocity (or e = A, if v is constant).I. 
§ 3. Such being the nature of the elastic force, it will not be difficult to prove 
that it increases exactly as the density of the medium. The proposition stands thus : 
if the number of molecules in a volume of the medium be doubled, the number of 
impacts that take place on a constant surface in a constant time will also be doubled, 
the velocity being unchanged. 
Suppose the number octupled, the mean distance is reduced to one-half. If they 
were equidistant and moving in one direction with the constant velocity, it is evident 
that eight times the previous number would pass the same imaginary plane in the 
same time, and if the plane were solid that eight times the previous number would 
impinge against it. Now, although all do not move in one direction, yet in both 
cases the same proportion of the whole must in each case do so. Whatever may be 
the density no preference can be assigned to one direction more than to another in 
the molecular movements; they must in every case be equally distributed in every 
* [The case is that where the particle (mass 1) and the plane (mass n) hoth reverse their velocities at 
impact. The conservation of vis viva is thereby secured, and the condition of momentum gives at once 
n[3 - v. — R.] 
MDCCCXCII.—A. C 
