Energy and Radiation Theory. 73 



i?ill lose all distinguishing characteristics, each group having 

 some of its members inside, and some outside atoms. We 

 must, during the long period of time referred to, compare the 

 average energies of the groups again and again *, and all that 

 the theorem of equipartition of energy is capable of telling 

 us (when properly applied) is that, in all but a relatively infini- 

 tesimal proportion of such comparisons, the average energies 

 will be equal. The theorem, however, does not deny that the 

 averages may be unequal in certain isolated cases, and such 

 cases are of course artificially brought into prominence when 

 we initially arrange the groups so that one contains only 

 electrons which are inside atoms, and the other contains only 

 electrons which are outside atoms. It is true that the above 

 argument would not apply if the electrons inside atoms 

 always remained inside ; such a case, however, would not 

 be one to which the theorem of equipartition would profess 

 to apply, for in this case it becomes denied at the outset that 

 the system can pass through all phases consistent with the 

 conservation of energy t- 



Again, if we consider a mixture of two gases, say oxygen 

 and hydrogen, and suppose that there are electrons inside 

 the atoms of each gas, and also free electrons, the theorem 

 of equipartition of energy, even if applicable, would not tell 

 us that vve could equate the average energy of the electrons 

 inside the hydrogen molecules to the average energy of the 

 electrons inside the oxygen molecules, or to the average 

 energy of the free electrons. It is only when we consider 

 the average in accordance w T ith the remarks above, or more 

 crudely when we choose two groups each of v>hich has a 

 large number of members drawn more or less indis- 

 criminately from the oxygen, hydrogen, and free electrons 

 that the theorem enables us to equate the averages, and in 

 this case of course the result could be predicted without 

 the theorem, and is indeed a conclusion of no interest. In 

 fact, the theorem in such cases tells us practically nothing 

 at all, and the same difficulty must always present itself 

 when we imagine all the matter concerned as specified by 

 coordinates which are of the same kind. Accord i no- to the 



* Strictly speaking-, the instants at which comparisons are made 

 should be so timed that if the corresponding representative points are 

 plotted in the generalized space, their distribution nlong the stream-line 

 will be the same as it would be in the case of a uniform distribution of 

 points throughout the whole space. 



t It must be remarked, however, that violation of this condition does 

 not necessarily invalidate equipartition, since equipartition holds for such 

 a large proportion of the generalized space. 



