CHAPTER VI. 



EXTENSION IN CONFIGURATION AND EXTENSION 

 IN VELOCITY. 



THE formulae relating to canonical ensembles in the closing 

 paragraphs of the last chapter suggest certain general notions 

 and principles, which we shall consider in this chapter, and 

 which are not at all limited in their application to the canon- 

 ical law of distribution.* 



We have seen in Chapter IV. that the nature of the distribu- 

 tion which we have called canonical is independent of the 

 system of coordinates by which it is described, being deter- 

 mined entirely by the modulus. It follows that the value 

 represented by the multiple integral (142), which is the frac- 

 tional part of the ensemble which lies within certain limiting 

 configurations, is independent of the system of coordinates, 

 being determined entirely by the limiting configurations with 

 the modulus. Now t|r, as we have already seen, represents 

 a value which is independent of the system of coordinates 

 by which it is defined. The same is evidently true of 

 typ by equation (140), and therefore, by (141), of ty g . 

 Hence the exponential factor in the multiple integral (142) 

 represents a value which is independent of the system of 

 coordinates. It follows that the value of a multiple integral 

 of the form 



^ ...dg n (148) 



* These notions and principles are in fact such as a more logical arrange- 

 ment of the subject would place in connection with those of Chapter I., to 

 which they are closely related. The strict requirements of logical order 

 have been sacrificed to the natural development of the subject, and very 

 elementary notions have been left until they have presented themselves in 

 the study of the leading problems. 



