Page:Elementary Principles in Statistical Mechanics (1902).djvu/84

60 or its equivalent an element of extension-in-velocity.

An extension-in-phase may always be regarded as an integral of elementary extensions-in-configuration multiplied each by an extension-in-velocity. This is evident from the formulae (151) and (152) which express an extension-in-phase, if we imagine the integrations relative to velocity to be first carried out.

The product of the two expressions for an element of extension-in-velocity (149) and (150) is evidently of the same dimensions as the product that is, as the $$n$$th power of energy, since every product of the form $$p_1\dot q_1$$ has the dimensions of energy. Therefore an extension-in-velocity has the dimensions of the square root of the $$n$$th power of energy. Again we see by (155) and (156) that the product of an extension-in-configuration and an extension-in-velocity have the dimensions of the $$n$$th power of energy multiplied by the $$n$$th power of time. Therefore an extension-in-configuration has the dimensions of the $$n$$th power of time multiplied by the square root of the $$n$$th power of energy.

To the notion of extension-in-configuration there attach themselves certain other notions analogous to those which have presented themselves in connection with the notion of extension-in-phase. The number of systems of any ensemble (whether distributed canonically or in any other manner) which are contained in an element of extension-in-configuration, divided by the numerical value of that element, may be called the density-in-configuration. That is, if a certain configuration is specified by the coördinates $$q_1\ldots q_n$$, and the number of systems of which the coördinates fall between the limits $$q_1$$ and $$q_1+dq_1$$,...$$q_n$$ and $$q_n+dq_n$$ is expressed by