Step a
Determination of the Number of Complexions of a given Physical Configuration of a Known Macrostate
We will, for simplicity's sake, consider here an ideal gas in a given macro-state and consisting of N-like, monatomic, molecules. By generalizing the meaning of our co-ordinates, the following presentation could be made equally applicable to the more general case of Physics contemplated under this heading.
Of course we must here have clearly in mind what is meant by the state of a gas. For this we may refer to [p. 10] (lines 13 to 24) and to [p. 19] (lines 8 to 24). The conditions there imposed are all fulfilled if we suppose the state given in such a way that we know: (1) The number of molecules in any macroscopically small space (volume element); and (2) the number of molecules which lie in a certain macroscopically small velocity region (soon to be more fully described). To have the Calculus of Probabilities applicable, each of the tiny regions contemplated under (1) and (2) must still contain a large number of molecules and their motions must besides have all the features of haphazard detailed on pp. [25], [26]; all this is necessary in order that the contemplated motions may possess all the characteristics of "elementary chaos."
Before proceeding further on our main line, we will define more fully what is meant by the two elementary regions in which lie respectively the molecules and their velocity ends. After this has been done we will, for convenience, combine these two regions into a fictitious elementary region, say, a six-dimensional one.
First there is the volume element
, in which any molecule having co-ordinates lying between
is located; this element can be conceived as a parallelopipedon whose edges are parallel to the co-ordinate axes; this is the simplest of the elementary regions here to be considered. To conceive of the elementary region
