or its equivalent
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(157)
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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
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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
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(158)
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