if n be the number of degrees of freedom of the molecule.
Thus, if this Boltzmann-Maxwell theorem be true, the specific heat of a gas will depend solely on the number of degrees of freedom of each of its molecules. For hard rigid bodies we should have n equal to 6, and hence γ = 1·333. Now the fact that this is not the value of γ for any of the known gases is a fundamental difficulty in the way of accepting the complete theory.
Boltzmann has called attention to the fact that if n be equal to five, then γ has the value 1·40. And this agrees fairly with the value found by experiment for air, oxygen, nitrogen, and various other gases. We will, however, return to this point shortly.
There is, perhaps, no result in the domain of physical science in recent years which has been more discussed than the two fundamental theorems of the molecular theory which we owe to Maxwell and to Boltzmann.
The two results in question are (1) the expression for the number of molecules which at any moment will have a given velocity, and (2) the proposition that the kinetic energy is ultimately equally divided among all the variables which determine the system.
With regard to (1) Maxwell showed that his error law was one possible condition of permanence. If at any moment the velocities are distributed according to the error law, that distribution will be a permanent one. He did not prove that such a distribution is the only one which can satisfy all the conditions of the problem.
The proof that this law is a necessary, as well as a sufficient, condition of permanence was first given by Boltzmann, for a single monatomic gas in 1872, for a mixture of such gases in 1886, and for a polyatomic gas in 1887. Other proofs have been given since by Watson and Burbury. It would be quite beyond the limits of this book to go into the question of the completeness or sufficiency of the proofs. The discussion of the question is still in progress.
The British Association Report for 1894 contains an important contribution to the question, in the shape of a report by Mr. G. H. Bryan, and the discussion he started at Oxford by reading this report has been continued in the pages of Nature and elsewhere since that time.
Mr. Bryan shows in the first place what may be the nature of the systems of molecules to which the results will apply, and discusses various points of difficulty in the proof.
The theorem in question, from which the result (1) follows as a simple deduction, has been thus stated by Dr. Larmor.[54]