36 CANONICAL DISTRIBUTION 



and the system B as one of an ensemble of systems of n 

 degrees of freedom distributed in phase with a probability- 

 coefficient 



which has the same modulus. Let q v . . .q m , p v . . . p m be the 

 coordinates and momenta of A, and q m+l , . . . q m+n , p m+l , . . . p m+n 

 those of . Now we may regard the systems A and B as 

 together forming a system 0, having m + n degrees of free- 

 dom, and the coordinates and momenta q^ . . . <?,+, p v . . . p m+n . 

 The probability that the phase of the system (7, as thus defined, 

 will fall within the limits 



dpi , . . . dp m+n , dq 1 , . . . dq m +n 



is evidently the product of the probabilities that the systems 

 A and B will each fall within the specified limits, viz., 



(94) 



We may therefore regard C as an undetermined system of an 

 ensemble distributed with the probability-coefficient 



(95) 



an ensemble which might be defined as formed by combining 

 each system of the first ensemble with each of the second. 

 But since e A + B is the energy of the whole system, and 

 ^ A and >/r B are constants, the probability-coefficient is of the 

 general form which we are considering, and the ensemble to 

 which it relates is in statistical equilibrium and is canonically 

 distributed. 



This result, however, so far as statistical equilibrium is 

 concerned, is rather nugatory, since conceiving of separate 

 systems as forming a single system does not create any in- 

 teraction between them, and if the systems combined belong to 

 ensembles in statistical equilibrium, to say that the ensemble 

 formed by such combinations as we have supposed is in statis- 

 tical equilibrium, is only to repeat the data in different 



