ON OUR KNOWLEDGE OF THERMODYNAMICS. 67 



The whole of Maxwell's demonstration, and most of the investigations 

 of Boltzmann/ Watson,^ and other writers on the same subject, are based 

 on the consideration of an infinitely large number of independent systems, 

 similar in every respect, whose co-ordinates and momenta at any instant 

 are distributed according to a fixed law, and the object is to find what 

 this distribution must be in order that it may be independent of the 

 time and unafiected by the motions of the systems. I cannot see that 

 these investigations anywhere assume that each individual system passes 

 through every possible phase. At each instant there must be some 

 systems in every possible phase ; but a distribution would obviously be 

 permanent — and very much so indeed — in which each system always 

 remained in the same phase, and never passed into any other phase. 



10. The assumption first confronts us when we attempt to pass from 

 the consideration of a large number of systems to that of a single system, 

 i.e., if, having investigated the result of averaging the distributions of 

 energy at a' given instant over the different systems, we wish to infer 

 similar properties for the corresponding time-averages for any one of the 

 systems. 



It is easy enough to suggest systems to which the assumption is 

 inapplicable. Most of the ' test cases ' which have been suggested as dis- 

 proving the law, and which will be considered later on, are instances of 

 such systems. Lord Eayleigh ^ has suggested as another instance an 

 elastic ball moving on a table having a circular boundary, at which it is 

 reflected. If, instead of taking a single particle. Lord Rayleigh had sup- 

 posed the table covered with such particles initially distributed uniformly 

 over its area, and projected in such a manner that at any point as many 

 particles were moving in one direction as in another, he would find these 

 same conditions satisfied at any subsequent time, and this is, to my mind, 

 all that Maxwell proves. 



11. It is far less easy to suggest any simple system which does satisfy 

 the assumption. The tracing point of a Lissajous' pendulum curve-tracer, 

 considered by Boltzmann,^ or in other words a particle whose equations 

 of motion are 



x + a^x=0, y-\-b"-y^Q, 



possesses when a, h are incommensurable the property of passing sooner 

 or later through every point within a certain rectangle, but it does not 

 possess the other necessary property of passing through any point in every 

 possible direction in succession. This may be easily seen for the simplest 

 case when a is nearly but not quite equal to b : here the path is nearly 

 elliptical, and there are only two possible directions at any point. Hence, 

 in order to satisfy the assumption, Boltzmann requires a thin elastic 

 cylinder to be placed perpendicularly to the plane of motion, so that the 

 particle may have its direction of motion changed each time it strikes and 

 rebounds from the cylinder. And this introduces collisions into the 

 problem. 



' ' TJeber die Eigenschaften monocyklischer unci anclerer damit verwandter 

 Systeme,' Journal fur die relne nnd angewandtc Mat he m at U/, xcviii. p. 68. 

 ' Analogien des zweiten Hauptsatzes,' ihid., c. pp. 206, 207, and other papers. 



- Kinetic Theory of Gases, new edition, p. 23. 



" Phil. Mag., April 1892, p 357. 



■* ' Ueber die mechanischen Analogien des zweiten Hauptsatzes der Thermo- 

 dynamik,' Journal fur die Mathematik, c. p. 203. 



p2 



