35. In the last chapter it was twice found convenient to represent the three velocity coordinates u, v, w of a molecule, by a point in space of which the coordinates referred to three rectangular axes were u, v, w. The principle involved is a useful one, capable of almost indefinite extension, and will be largely used both in the present chapter and elsewhere in the book.

The space of nature possesses three dimensions, but just as it is open for us to represent any two coordinates in an imaginary space of only two dimensions, so in the same way we may represent any four coordinates in an imaginary space of four dimensions. Similarly if a dynamical system is specified by any number n of coordinates, we can represent these coordinates in a space of n dimensions, and the various points in this space will correspond to the various configurations of the dynamical system.

In the present chapter, we attempt to find the law of distribution of velocities by a method which consists essentially in regarding the whole gas as a single dynamical system, and in representing its coordinates in a single imaginary space of the appropriate number of dimensions.

Let us suppose that the gas consists of a great number N of exactly similar molecules, enclosed in a vessel of volume Ω.