J. 



CANONICAL DISTRIBUTION. 33 



cient of probability, whether the case is one of equilibrium 

 or not. These are: that P should be single-valued, and 

 neither negative nor imaginary for any phase, and that ex- 

 pressed by equation (46), viz., 



all 



JP4>...- <*? = !. (89) 



phases 



These considerations exclude 



P = e X constant, 



as well as 



P = constant, 



as cases to be considered. 



The distribution represented by 



(90) 



or 



where and i/r are constants, and % positive, seems to repre- 

 sent the most simple case conceivable, since it has the property 

 that when the system consists of parts with separate energies, 

 the laws of the distribution in phase of the separate parts are 

 of the same nature, a property which enormously simplifies 

 the discussion, and is the foundation of extremely important 

 relations to thermodynamics. The case is not rendered less 

 simple by the divisor , (a quantity of the same dimensions as 

 e,) but the reverse, since it makes the distribution independent 

 of the units employed. The negative sign of e is required by 

 (89), which determines also the value of ^ for any given 

 , viz., 



all f 



~ 



=f. . .f 



e dp,... dq n . (92) 



phases 



When an ensemble of systems is distributed in phase in the 

 manner described, i. e.^ when the index of probability is a 



3 



