ox OUR KNOWLEDGE OF THERMODYNAMICS. 93 



babilities of all energies are equal for any one molecule. If m is the 

 mass of a molecule, the most probable number of molecules with speeds 

 between u and u + du is 



Ce-''"""= mudu (.51) 



This is the Boltzmann-Maxwell distribution of speed for a system of 

 monatomic molecules moving in one plane. 



53. To obtain the Boltzmann-Maxwell distribution for molecules 

 moving in three dimensions, Boltzmann finds it necessary to make a 

 different assumption with regard to the «;;riori probabilities. He assumes, 

 in fact, that if u, v, to be the velocities along the axes of co-ordinates, all 

 values of u, v, w are a priori equally probable. The problem of deter- 

 mining the a ])osteriori most probable distribution therefore reduces to 

 finding the minimum of 



\\\f^<^gf • du dv dio= —n SM^^o?,e . . . (.52) 



subject to the conditions 



T=l7n{ {{u'' + v'^ + iv'-)dudvdw . . . (54) 



For the general case of molecules with r degrees of freedom in a field 

 of force, he assumes that all values of the co-ordinates and momenta p are 

 « priori equally probable. 



This is the assumption that would be fulfilled if the values were 

 selected by drawings from an urn containing tickets, each ticket having a 

 set of values of ;j, . . . q,. inscribed on it, and the number of tickets in 

 which these values lie between the limits dp^ . . . dq^ being measured by 

 the product of the multiple differential dp^ . . . dq, into a constant. In 

 the case of a mixture of gases there would have to be a number of urns 

 equal to the number of gases, and the number of tickets drawn from each 

 urn would have to equal the number of molecules of the gas in question in 

 the mixture. 



The final result is that the a 2>osteriori most probable distribution is 

 that for which the function 



-il=^[f\ogfdp, . . . dq,. . . (55) 



is a minimum, 2 referring to the different kinds of moleiules in a mixture 

 of several gases. 



The expx-ession —H differs by a constant from JBoltzmann's Minimum 

 Ftinction. 



We have seen in § 43 that, when there are collisions between the 

 molecules, this function always tends to a minimum until the Boltzmann- 

 Maxwell distribution is attained, and the present investigation therefore 

 shows that the gas tends to pass from distributions of lesser probability 

 to distributions of greater probability, until it attains the most probable 

 distribution of all — namely, the Boltzmann-Maxwell distribution. 



Finally, Boltzmann proves that the function ii is proportional to the 

 entropy (plus a constant), thus affording a verification of the theorem that 

 the entropy of a system tends to a minimum. The identification of Boltz- 



