xii PREFACE. 



with which we experiment, if distributed canonically, would 

 therefore appear to human observation as an ensemble of 

 systems in which all have the same energy. 



We meet with other quantities, in the development of the 

 subject, which, when the number of degrees of freedom is 

 very great, coincide sensibly with the modulus, and with the 

 average index of probability, taken negatively, in a canonical 

 ensemble, and which, therefore, may also be regarded as cor- 

 responding to temperature and entropy. The correspondence 

 is however imperfect, when the number of degrees of freedom 

 is not very great, and there is nothing to recommend these 

 quantities except that in definition they may be regarded as 

 more simple than those which have been mentioned. In 

 Chapter XIV, this subject of thermodynamic analogies is 

 discussed somewhat at length. 



Finally, in Chapter XV, we consider the modification of 

 the preceding results which is necessary when we consider 

 systems composed of a number of entirely similar particles, 

 or, it may be, of a number of particles of several kinds, all of 

 each kind being entirely similar to each other, and when one 

 of the variations to be considered is that of the numbers of 

 the particles of the various kinds which are contained in a 

 system. This supposition would naturally have been intro- 

 duced earlier, if our object had been simply the expression of 

 the laws of nature. It seemed desirable, however, to separate 

 sharply the purely thermodynamic laws from those special 

 modifications which belong rather to the theoiy of the prop- 

 erties of matter. 



J. W. G. 



NEW HAVEN, December, 1901. 



