AND EXTENSION IN VELOCITY. 61 



D q will be the density-in-configuration. And if we set 



*=ip ( 159 ) 



where N denotes, as usual, the total number of systems in the 

 ensemble, the probability that an unspecified system of the 

 ensemble will fall within the given limits of configuration, is 

 expressed by 



e^dq t . . . dq n . (160) 



We may call &* the coefficient of probability of the, configura- 

 tion, and t] q the index of probability of the configuration. 



The fractional part of the whole number of systems which 

 are within any given limits of configuration will be expressed 

 by the multiple integral 



J. 



. . . dg n . (161) 



The value of this integral (taken within any given configura- 

 tions) is therefore independent of the system of coordinates 

 which is used. Since the same has been proved of the same 

 integral without the factor e* q , it follows that the values of 

 7) q and D q for a given configuration in a given ensemble are 

 independent of the system of coordinates which is used. 



The notion of extension-in-velocity relates to systems hav- 

 ing the same configuration.* If an ensemble is distributed 

 both in configuration and in velocity, we may confine our 

 attention to those systems which are contained within certain 

 infinitesimal limits of configuration, and compare the whole 

 number of such systems with those which are also contained 



* Except in some simple cases, such as a system of material points, we 

 cannot compare velocities in one configuration with velocities in another, and 

 speak of their identity or difference except in a sense entirely artificial. We 

 may indeed say that we call the velocities in one configuration the same as 

 those in another when the quantities q lt ...q n have the same values in the 

 two cases. But this signifies nothing until the system of coordinates has 

 been defined. We might identify the velocities in the two cases which make 

 the quantities pi,...p n the same in each. This again would signify nothing 

 independently of the system of coordinates employed. 



