Page:Elementary Principles in Statistical Mechanics (1902).djvu/89

But the systems which initially had velocities satisfying the equation (172) will after the interval $\delta t$ have configurations satisfying equation (177). Therefore the extension-in-configuration represented by the last integral is that which belongs to the systems which originally had the extension-in-velocity represented by the integral (171).

Since the quantities which we have called extensions-in-phase, extensions-in-configuration, and extensions-in-velocity are independent of the nature of the system of coördinates used in their definitions, it is natural to seek definitions which shall be independent of the use of any coördinates. It will be sufficient to give the following definitions without formal proof of their equivalence with those given above, since they are less convenient for use than those founded on systems of coördinates, and since we shall in fact have no occasion to use them.

We commence with the definition of extension-in- velocity. We may imagine $n$ independent velocities, $V_{1},\ldots V_{n}$ of which a system in a given configuration is capable. We may conceive of the system as having a certain velocity $V_{0}$ combined with a part of each of these velocities $V_{1}\ldots V_{n}$. By a part of $V_{1}$ is meant a velocity of the same nature as $V_{1}$ but in amount being anything between zero and $V_{1}$. Now all the velocities which may be thus described may be regarded as forming or lying in a certain extension of which we desire a measure. The case is greatly simplified if we suppose that certain relations exist between the velocities $V_{1},\ldots V_{n}$, viz: that the kinetic energy due to any two of these velocities combined is the sum of the kinetic energies due to the velocities separately. In this case the extension-in-motion is the square root of the product of the doubled kinetic energies due to the $n$ velocities $V_{1},\ldots V_{n}$ taken separately.

The more general case may be reduced to this simpler case as follows. The velocity $V_{2}$ may always be regarded as composed of two velocities $V_{2}{}'$ and $V_{2}{}''$, of which $V_{2}{}'$ is of the same nature as $V_{1}$, (it may be more or less in amount, or opposite in sign,) while $V_{2}{}''$ satisfies the relation that the