62 EXTENSION IN CONFIGURATION 



within certain infinitesimal limits of velocity. The second 

 of these numbers divided by the first expresses the probability 

 that a system which is only specified as falling within the in- 

 finitesimal limits of configuration shall also fall within the 

 infinitesimal limits of velocity. If the limits with respect to 

 velocity are expressed by the condition that the momenta 

 shall fall between the limits p 1 and p 1 + dp l , . . . p n and 

 Pn + dpm the extension-in-velocity within those limits will be 



. . . dp n , 

 and we may express the probability in question by 



e^\^d Pl . . . dp n . (162) 



This may be regarded as defining rj p . 



The probability that a system which is only specified as 

 having a configuration within certain infinitesimal limits shall 

 also fall within any given limits of velocity will be expressed 

 by the multiple integral 



h . . . dp n , (163) 



or its equivalent 



J 1 . . .J**4Mb . . . dg n , (164) 



taken within the given limits. 



It follows that the probability that the system will fall 

 within the limits of velocity, ^ and ^ + dq 19 . . . q n and 

 2 + dq* is expressed by 



e^^d^^.d^. (165) 



The value of the integrals (163), (164) is independent of 

 the system of coordinates and momenta which is used, as is 

 also the value of the same integrals without the factor 

 e 1 ?; therefore the value of TJ P must be independent of the 

 system of coordinates and momenta. We may call e 1 ? the 

 coefficient of probability of velocity, and tj p the index of proba- 

 bility of velocity. 



