64 EXTENSION IN CONFIGURATION 



In the general case, the notions of extension-in-configuration 

 and extension-in-velocity may be connected as follows. 



If an ensemble of similar systems of n degrees of freedom 

 have the same configuration at a given instant, but are distrib- 

 uted throughout any finite extension-in-velocity, the same 

 ensemble after an infinitesimal interval of time St will be 

 distributed throughout an extension in configuration equal to 

 its original extension-in-velocity multiplied by $t n . 



In demonstrating this theorem, we shall write q^ . . . q n f for 

 the initial values of the coordinates. The final values will 

 evidently be connected with the initial by the equations 



Now the original extension-in-velocity is by definition repre- 

 sented by the integral 



J. . ,JV4i - <&, (171) 



where the limits may be expressed by an equation of the form 

 F(j ll ...^) = Q. (172) 



The same integral multiplied by the constant St* may be 

 written 



J. . . jVd&ft), . . . %&), (173) 



and the limits may be written 



(It will be observed that St as well as A^ is constant in the 

 integrations.) Now this integral is identically equal to 



f. . ./A,* d(q, - <?/) . . . d(q, . . . ft,'), (175) 



or its equivalent 



AM. * (176) 



f. -/ 



with limits expressed by the equation 



/ (ft -<?/, 2.- 2,.') =0. (177) 



