to each sphere according to any law subject to the conditions that
and that
given constant. (2) Another person, in complete ignorance of the velocities so assigned, scatters the spheres at haphazard throughout
. And they start from the initial positions so assigned by (2) with the velocities assigned to them respectively by (1)."
The system thus synthetically constructed would without doubt, at the start be "molekular-ungeordnet"—in fact, it is as near an approach to chaos as is possible in an imperfect world. But there is reason to doubt if it would continue to be thus "molekular-ungeordnet." For the distribution of velocities is according to any law consistent with the above-mentioned conditions and some such laws would lead to results hostile to the Second Law, and then we may safely say such laws of velocity distribution would never occur in Nature and would therefore belong to the cases which have been specially excepted.
Now there are mechanical systems which possess the entropy property and it has been truly said that the Second Law and irreversibility do not depend on any special peculiarity of heat motion, but only on the statistical property of a system possessing an extraordinary number of degrees of freedom. In this sense Professor J. W. GIBBS treated Mechanics statistically and showed that then the properties of temperature and entropy resulted. This matter has already been touched upon, but as numerous degrees of freedom is a feature of the "elementary chaos" under consideration it deserves repetition here and more than a passing mention.
Illustration of Degrees of Freedom. Refer a body's motion to three axes,