SYSTEMS COMPOSED OF MOLECULES. 197 



obtained by combining the two original ensembles. The 

 difference between the ensemble which would be in statistical 

 equilibrium, and that obtained by combining the two original 

 ensembles may be diminished without limit, while it is still 

 possible for particles to pass from one system to another. In - 

 this sense we may say that the ensemble formed by combining 

 the two given ensembles may still be regarded as in a state of 

 (approximate) statistical equilibrium with respect to generic 

 phases, when it has been made possible for particles to pass 

 between the systems combined, and when statistical equilibrium 

 for specific phases has therefore entirely ceased to exist, and 

 when the equilibrium for generic phases would also have 

 entirely ceased to exist, if the given ensembles had not been 

 canonically distributed, with respect to generic phases, with 

 the same values of @ and fi v . . . p h . 



It is evident also that considerations of this kind will apply j 

 separately to the several kinds of particles. We may diminish ' 

 the energy in the space forming the diaphragm for one kind of 

 particle and not for another. This is the mathematical ex- 

 pression for a " semipermeable" diaphragm. The condition 

 necessary for statistical equilibrium where the diaphragm is 

 permeable only to particles to which the suffix ( ) x relates 

 will be fulfilled when /^ and have the same values in the 

 two ensembles, although the other coefficients /* 2 , etc., may be 

 different. 



This important property of grand ensembles with canonical 

 distribution will supply the motive for a more particular ex- 

 amination of the nature of such ensembles, and especially of 

 the comparative numbers of systems in the several petit en- 

 sembles which make up a grand ensemble, and of the average 

 values in the grand ensemble of some of the most important 

 quantities, and of the average squares of the deviations from 

 these average values. 



The probability that a system taken at random from a 

 grand ensemble canonically distributed will have exactly 

 i/j, . . . v h particles of the various kinds is expressed by the 

 multiple integral 



