118 EEPORT — 1891. 



where 



,=^, H=-T=-2'^, s=J^=-^=2av . (76) 

 ^ 2a on ' clq m 



and — 9H/3a is the pressni'e on either wall. 



This system is strictly monocyclic. 



Boltzmann modifies this example slightly by considering the case of a 

 mass m formed of minute particles contained in a rectangular box, whose 

 sides are a, h, c, the directions of motion being parallel to the face (ah) 

 and inclined to the edges a at an angle=D. Taking a, h, and v as 

 variable, we have 



mv^ /" (In db\ 



-H=T=-2-, dQ=7nvdv + mv' I sm-B -^+ cos- D~^ J (77) 



and to put the last equation into Helmholtz's form we must assume 



But the kinetic energy is no longer an integrating divisor of (ZQ if we 

 suppose the angle T> variable. It is not hard to explain why this case 

 differs from the others considered by Boltzmann. The angle D cannot 

 be considered as a controllable coordinate of the system, for it can only 

 be varied by acting on all the molecules individually. Moreover, it is 

 not a speed-coordinate, so that Helmholtz's methods are no longer applic- 

 able. The effect of slightly rotating the box would be not merely to 

 l^roduce an alteration in the angle of incidence D, but to alter the charac- 

 ter of the motion entirely, for the particles which are about to impinge 

 on the face ac would be differently affected from those about to impinge 

 on the face be. 



Boltzmann follows up these simple examples by a perfectly general 

 investigation based on Maxwell's theorem, from which it appears that 

 any system which conforms to the Boltzmann-Maxwell doctrine possesses 

 monocyclic properties analogous to those found by Helmholtz. The 

 results obtained by Boltzmann do not hold good, except in the particular 

 cases when Maxwell's theorem is valid. Two cases are considered — 

 that in which all coordinates of the system are independent, and that 

 in which certain coordinates are connected by invariable relations. The 

 arguments employed by Boltzmann in discussing the latter case appear 

 wanting in rigour, thus rendering the result liable to further objections. 

 The remainder of the paper is chiefly taken up with a discussion of the 

 models referred to in our second section. 



48. Application of Statistical Methods to Irreversible Plienoviena. — In a 

 recent note ' Mr. E. P. CulverAvell has called attention to the principal 

 difficulties attending the explanation of irreversibility on the hypotheses 

 of the kinetic theory of gases. The general purport of his remarks may 

 be summarised as follows : — 



(i.) Although the distribution of energy when a gas has assumed the 

 Boltzmann configuration (or, as Tait calls it, the ' special state ') has 

 been investigated, it has never been proved that a gas does actually tend 

 towards this ' special state.' 



• 'Note on Boltzmann's Kinetic Theory of Gases, and on Sir W. Thom.son's 

 Address to Section A (1884),' Fkil. Mag. 1890, vol. xxx. p. 95. 



