16 CONSERVATION OF 



It is the relative numbers of systems which fall within dif- 

 ferent limits, rather than the absolute numbers, with which we 

 are most concerned. It is indeed only with regard to relative 

 numbers that such discussions as the preceding will apply 

 with literal precision, since the nature of our reasoning implies 

 that the number of systems in the smallest element of space 

 which we consider is very great. This is evidently inconsist- 

 ent with a finite value of the total number of systems, or of 

 the density-in-phase. Now if the value of D is infinite, we 

 cannot speak of any definite number of systems within any 

 finite limits, since all such numbers are infinite. But the 

 ratios of these infinite numbers may be perfectly definite. If 

 we write -ZVfor the total number of systems, and set 



r = %. (38) 



P may remain finite, when JV* and D become infinite. The 

 integral 



" * ... dq n (39) 



taken within any given limits, will evidently express the ratio 

 of the number of systems falling within those limits to the 

 whole number of systems. This is the same thing as the 

 probability that an unspecified system of the ensemble (i. e. 

 one of which we only know that it belongs to the ensemble) 

 will lie within the given limits. The product 



Pd Pl ...dq n (40) 



expresses the probability that an unspecified system of the 

 ensemble will be found in the element of extension-in-phase 

 dpi . . . dq n . We shall call P the coefficient of probability of the 

 phase considered. Its natural logarithm we shall call the 

 index of probability of the phase, and denote it by the letter 77. 

 If we substitute NP and Ne 1 for D in equation (19), we get 



and 



