60 EXTENSION IN CONFIGURATION 



or its equivalent 



. . d, (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 



Pi- ' -PnVl --it 



that is, as the nth power of energy, since every product of the 

 form p l q 1 has the dimensions of energy. Therefore an exten- 

 sion-in-velocity has the dimensions of the square root of the 

 nth 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 nth power of energy 

 multiplied by the nth power of time. Therefore an extension- 

 in-configuration has the dimensions of the nth power of time 

 multiplied by the square root of the nth 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 ex- 

 tension-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-configura- 

 tion, divided by the numerical value of that element, may be 

 called the density-in-configuration. That is, if a certain con- 

 figuration is specified by the coordinates q 1 . . . q n , and the 

 number of systems of which the coordinates fall between the 

 limits q 1 and q l + dq l , . . . q n and q n + dq n is expressed by 



D.A^Zi *2n, (158) 



