26 THE PHYSICAL SIGNIFICANCE OF ENTROPY 



for its legitimate application. Of course the statement of these 

 features will vary with the mechanical or non-mechanical character 

 of the problem to which it is applied. As we are here dealing 

 mainly with the former, we will limit ourselves to its features : (a) 

 The elements dealt with must be very numerous, strictly speaking, 

 infinite; (b) as a phase of (a) we may say also that when we 

 speak of the probability of a state we express the thought that 

 it can be realized in many different ways; (c) when we speak 

 of the relative directions o-f a pair of molecules all possible direc- 

 tions must be considered; (d) we must so weight the elements 

 say, in (a), (6), and (c) that they are equally likely; (e) every one 

 of the entering elements must possess constituents of which each 

 individual is independent of every other; for instance, (/) in a gas 

 the place where a molecule collided must be independent of the 

 place where it collided before. In our physical problem all of 

 these features are not always realized; for instance, the number 

 of particles of gas are only finite instead of being infinite; 

 again, all relative velocities after collision of a pair of molecules 

 are not equally likely; BOLTZMANN and BURBURY provide for these 

 shortcomings by very truly asserting that in actual cases we are 

 not dealing with isolated systems, that the surrounding walls 

 are not impervious to external influences, and that the latter 

 come at haphazard without regard to internal state of the system 

 at the time, thus renewing and maintaining the desired state of 

 haphazard. 



Methods. This Calculus works largely by the determination 

 of averages and its results must be interpreted accordingly. 

 Moreover, for the present we will take a popular, practical view 

 of these results and consider a very great improbability as equiva- 

 lent to an impossibility. Numerical computations are essential 

 in most uses of this Calculus, but here they will be entirely omitted. 



