Febkuaby 2, 1906.] 



SCIENCE. 



165 



we have certain knowledge, and can not 

 without proof be admitted as likely to ob- 

 tain in other systems, about which less is 

 known. In the method employed by Jeans 

 the application of the calculus of probabili- 

 ties is made in a different manner, and does 

 not necessitate the introduction of the hy- 

 pothesis of Maxwell and Boltzmann. It 

 seems to me that, in this last form of the 

 theory, the difficulties which have en- 

 vironed the subject have at last been mas- 

 tered. 



In respect to the third question, that 

 concerning the experimental evidence for 

 the truth of the theorem, it is well known 

 that, in general, Boyle's law follows as a 

 consequence of the general principles of the 

 kinetic theory, that Gay-Lussac's law is an 

 immediate consequence of a relation plau- 

 sibly assumed between temperature and the 

 kinetic energy of the molecule, that the 

 motion of the radiometer and the laws of 

 transpiration and many other properties of 

 gases can be deduced from the general 

 theory ; and, in particular, that Avogadro 's 

 law follows from the simplest form of the 

 theorem of equipartition. But further 

 proof of this theorem in its general form 

 is still needed. Such proof as we have will 

 be discussed later in this address. 



Let us consider a little more fully the 

 proofs of the theorem of equipartition. It 

 was first enunciated in 1845, in a paper 

 presented by Waterston to the Royal So- 

 ciety, but this paper was not published at 

 that time, and Maxwell's paper of 1859 

 first brought the theorem to the attention 

 of the scientific world. In that paper Max- 

 well undertook to prove that when two 

 systems of molecules move in the same ves- 

 sel the mean vis viva of each particle will 

 become the same in the two systems. 



The proof of this proposition by Max- 

 well is the one still commonly employed in 

 elementary expositions of the kinetic theory 

 of gases. As applied first to a single gas, 



he considers the molecules or particles of 

 the gas as elastic spheres, and represents 

 the average number of particles which have 

 a velocity in one direction lying between 

 two very near limits as a function of that 

 velocity. To represent the average number 

 of particles which have a velocity in either 

 of the other two rectangular directions 

 lying between certain near limits, he uses 

 the same function, and he then supposes 

 that the three velocities thus used are inde- 

 pendent of each other, so that the average 

 number of particles which possess all three 

 velocities at once will be given by the 

 product of three independent probabilities. 

 Since this number depends only on the 

 relative motions of the particles, and not 

 on the particular directions in which the 

 coordinate axes have been drawn, it may 

 also be represented by a function of the 

 resultant velocity of the particles, or by a 

 function of the sum of the squares of the 

 component velocities. Equating these two 

 expressions, a functional equation is ob- 

 tained, the solution of which leads to the 

 well-known exponential law of the distribu- 

 tion of velocities among the molecules. 



By extending the method just described 

 to the consideration of the relative motion 

 of the particles of two gases. Maxwell pro- 

 ceeded to show that the probable number 

 of particles, whose velocities differ by a 

 certain amount, is expressed by the same 

 exponential function as that already ob- 

 tained ; and he shows further that the prob- 

 able mean relative velocity is the square 

 root of the sum of the squares of the mean 

 velocities in the two systems. 



On the basis of this proposition. Max- 

 well proves that the average kinetic energy 

 of the molecules of two or more gases, 

 when they are mixed, will be the same for 

 each. To do this he shows simply that the 

 difference of the mean kinetic energies of 

 the molecules of two gases will be dimin- 

 ished by collision, so that it is only neces- 



