10. The mathematical difficulties of the subject commence when we attempt to discuss the law according to which the velocities of the molecules are grouped about their mean value. We are of course at liberty to consider an imaginary gas in which the velocities are grouped at the outset according to any law we please, but in general every collision which occurs will tend to change this law. The problem before us is to investigate whether there is any law which remains, on the whole, unchanged by collisions; and if so whether the velocities of the molecules of a gas, starting from some arbitrarily chosen law, will tend after a sufficient time to obey some definite law which is independent of the particular law from which the gas started.

There are two totally distinct methods of attacking these problems, and these are given in this chapter and the next, the relation between them being discussed in Chapter IV. The present chapter contains the classical method of which the development is due mainly to Clerk Maxwell and Boltzmann (see § 60 below).

The definition of Density.

11. There is no difficulty in defining the density of a continuous substance. If we take a small volume v, enclosing a given point P, and denote by m the mass of matter contained within this volume, then the assumption of continuity ensures that as the volume v shrinks until it is of infinitesimal size, while still enclosing the point P, then the ratio m/v will approach a definite limit ρ, and we define the density at the point P as being the value of the limit ρ.