. This parallelopiped can be regarded as a constructed volume within which the velocity end must lie. We have therefore here two elementary volumes
and
, which are independent of each other. Now remembering that the probability of any properly endowed molecule being found in one of these volumes is in each case equal to the number of molecules belonging or corresponding to the volume considered. Assuming, for the moment, an equal distribution of molecules and velocities throughout the whole volume, we may say that the number of molecules occurring, in each of the said elementary volumes, is proportional to their respective sizes; this is here equivalent to saying that the probability of any molecule thus occurring in said elementary volumes is proportional to their respective sizes. Having stated the probability of each contemplated occurrence, we can now say the probability of these two events concurring is equal to the product of the probabilities of said two separate occurrences. Moreover, as the probability of each occurrence is proportional to the size of its own elementary volume, the product of said probabilities will likewise be proportional to the product
of the two elementary volumes. Here
can be regarded as a sort of fictitious volume or region, constructed, say, in a six-dimensional space.[20]
The extent of such an elementary region is very minute in comparison with the total space under consideration, but still it must be conceived as sufficiently large to embrace many molecules, otherwise its state would not be one of "elementary chaos." On account of the equivalence here of probability and number of concurring molecules, we may for the present say that the number of the latter is proportional to the magnitude of this elementary region