ON OUR KNOWLEDGE OF THERMODYNAMICS. 95 



chance that the values oi x^, X2 . . . shall lie between c, and Ci+dciy 

 C2 and C2 + dc.2, &c., is proportional to an expression of the form 



_IIS 



e -T dci dc2 . . . dc„ .... (58) 



where S is a certain homogeneous quadratic function of the c's, and T a, 

 constant. This result, which for a single variable leads to the well-known 

 error-law, is independent of the original law of distribution of the inci-e- 

 raents, provided that positive and negative values of these increments are 

 equally probable. 



Taking S as proportional to the kinetic energy of a system, and sup- 

 posing the number of such systems to be very gi-eat, Burbury next sliows 

 that if a redisti'ibution of S among the systems is effected in a certain 

 way, the ultimate result will be the Boltzmann-Maxwell distribution, and 

 this will remain unaffected by any further redistribution. The method of 

 redistribution is such that energy is conserved in the final result, but not 

 in the intermediate processes, and Burbury suggests that the process of 

 redistribution of energy between the molecules may be effected by waves 

 transmitted through the ether. The proof requires us to assume that these 

 waves satisfy the principle of superposition, otherwise the law cannot be 

 permanent. The author, however, claims that the method is applicable to 

 systems in which no group of molecules is ever free from the action of 

 other parts of the system, and for which those proofs of the Boltzmann- 

 Maxwell Law treated in Sections I., II. of this Report fail. 



Burbury then finds the expression for Boltzmann's minimum function, 

 and calling this B he verifies that the entropy of the system is equal 

 to — 2B/tc (plus a constant). The whole ti'eatment is very powerful 

 and suggestive, and the paper opens up a wide field for discussion and 

 speculation. 



56. The assumption in the first place that each molecule is capable of 

 assuming only a discrete instead of a continuous series of different states, 

 the number of these states being made infinite in the limit, forms the basis 

 of Boltzmann's proof of his Minimum Theorem for polyatomic gas-mole- 

 cules.' Natanson,^ taking Boltzmann's starting-point of a number of 

 systems whose energies can only have one of a series of discrete values 



f, 2f, . . . pe 



and employing equations (43) (44) above, has worked out the final dis- 

 tribution of energy among the molecules on the supposition that inter- 

 change of energy takes place by collisions, and he has also determined 

 the rate at which the system approaches the Boltzmann-Maxwell 

 distribution. He gives a complete solution of the problem for the 

 particular case where ;J=3, and shows that only analytical difiiculties 

 prevent the method from being applied to higher values of p. When, 

 however, p is made infinite, the results agree with those found by Boltz- 

 mann and by Tait (§ 44). 



' ' Neuer Peweis zweier Siitze,' Sitzher. tier Ic. Wiener Altad., scv. (ii.)," Jan. 1S87, 

 p. 153. 



^ ' Ueber rlie Geschwindigkeit mit welcher Gase rlen Maxwell'schen Zustand 

 erreichen,' Anvale7i der Phijnh und Chemic, xxxiv. (188S). 



