300 On Boltzmann's Theorem on the average Distribution 



an abstract ; and, further, my notice contains one on two new 

 things, amongst which a remark on Watson's excellent book 

 may be interesting to English physicists. 



With highest esteem, 



Yours <fec, 



Boltzmann. 



Maxwell (Camb. Phil. Trans, vol. xii. part 3, pp. 547- 

 570, 1879) shows that this theorem may be easily proved 

 by means of Hamilton's principle. The theorem is also 

 extended, since it is shown to hold good for any systems 

 determined by generalized coordinates, if only they satisfy the 

 principle of conservation of energy. There is a difference in 

 method between Maxwell and Boltzmann, inasmuch as Boltz- 

 mann measures the probability of a condition by the time 

 during which the system possesses this condition on the ave- 

 rage, whereas Maxwell considers innumerable similarly con- 

 stituted systems with all possible initial conditions. The ratio 

 of the number of systems which are in that condition to the 

 total number of systems determines the probability in ques- 

 tion. In conclusion. Maxwell finds, further, that also for any 

 unstable system of very many. atoms in rotation under the 

 action of no external forces the mean energy of internal mo- 

 tion is the same for each atom, and that a mixture of gases in 

 a rotating tube behaves exactly as if each gas were present by 

 itself. 



Maxwell's proof mentioned above is as follows : — Let there 

 be given any system S obeying the principle of energy. Let 

 its configuration be determined by n generalized coordinates 

 rj l . . . q n ; let the corresponding momenta be p x . . ,p n . (For the 

 sake of clearness I will take occasionally the simplest example, 

 a system of material points acted on by any forces. q x . . . q„ will 

 then denote rectangular coordinates, p x ...p n the products of 

 the component velocity into the corresponding masses.) 



Let the law of the forces acting in the first system be such 

 that the potential energy Y is a given function of the coordinates. 

 Then the motion of the system is completely determined when 

 we know the values q\ . . . p' n of the coordinates and momenta 

 at the commencement of motion and the time t which has 

 elapsed. (In the example this means that the coordinates and 

 component velocities at the commencement of motion must be 

 known.) It is then most natural to take the 2n+ 1 quantities 

 '/\ • - •[''», t as so-called independent variables. Since the law 

 of action of the forces is given, all other quantities relating to 

 the motion (e. a. the values of the coordinates and momenta 

 after the lapse of the time t, which Maxwell denotes by 



