34 CANONICAL DISTRIBUTION 



linear function of the energy, we shall say that the ensemble is 

 canonically distributed, and shall call the divisor of the energy 

 () the modulus of distribution. 



The fractional part of an ensemble canonically distributed 

 which lies within any given limits of phase is therefore repre- 

 sented by the multiple integral 



9 dp l . . . dq n (93) 



taken within those limits. We may express the same thing 

 by saying that the multiple integral expresses the probability 

 that an unspecified system of the ensemble (i. e., one of 

 which we only know that it belongs to the ensemble) falls 

 within the given limits. 



Since the value of a multiple integral of the form (23) 

 (which we have called an extension-in-phase) bounded by any 

 given phases is independent of the system of coordinates by 

 which it is evaluated, the same must be true of the multiple 

 integral in (92), as appears at once if we divide up this 

 integral into parts so small that the exponential factor may be 

 regarded as constant in each. The value of ^r is therefore in- 

 dependent of the system of coordinates employed. 



It is evident that ty might be defined as the energy for 

 which the coefficient of probability of phase has the value 

 unity. Since however this coefficient has the dimensions of 

 the inverse nth power of the product of energy and time,* 

 the energy represented by -\Jr is not independent of the units 

 of energy and time. But when these units have been chosen, 

 the definition of ^ will involve the same arbitrary constant as 

 e, so that, while in any given case the numerical values of 

 ^r or e will be entirely indefinite until the zero of energy has 

 also been fixed for the system considered, the difference ty e 

 will represent a perfectly definite amount of energy, which is 

 entirely independent of the zero of energy which we may 

 choose to adopt. 



* See Chapter I, p. 19. 



