58 EXTENSION IN CONFIGURATION 



is independent of the system of coordinates which is employed 

 for its evaluation, as will appear at once, if we suppose the 

 multiple integral to be broken up into parts so small that 

 the exponential factor may be regarded as constant in each. 

 In the same way the formulae (144) and (145) which express 

 the probability that a system (in a canonical ensemble) of given 

 configuration will fall within certain limits of velocity, show 

 that multiple integrals of the form 



(149) 



or * **&. 1* (150) 



relating to velocities possible for a given configuration, when 

 the limits are formed by given velocities, have values inde- 

 pendent of the system of coordinates employed. 



These relations may easily be verified directly. It has al- 

 ready been proved that 



d(P l9 . . . P.) <%i . . . q n ) d(q l9 ...q n ) 



..-) d(Q l9 ...Q n ) 



where q l , . . . q^ft , . . .p n and Q l , . . . Q n9 P 1 , . . . P n are two 

 systems of coordinates and momenta.* It follows that 



i> 



= r 



J 



* See equation (29). 



