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

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60
EXTENSION IN CONFIGURATION

or its equivalent

(157)
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 th power of energy, since every product of the form has the dimensions of energy. Therefore an extension-in-velocity has the dimensions of the square root of the 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 th power of energy multiplied by the th power of time. Therefore an extension-in-configuration has the dimensions of the th power of time multiplied by the square root of the 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 , and the number of systems of which the coördinates fall between the limits and ,... and is expressed by

(158)