ON OUR KNOWLEDGE OF THERMODYNAMICS. 85 



molecules, Hj the mass of a molecule (so that vij-=fjjn^, ^j a function of T 

 and P, but not of rij, and E, a constant. The investigation is instructive 

 and suggestive ; unfortunately, however, the method of demonstration does 

 not appear to be ' perfectly reversible,' but it is much to be hoped that 

 Natanson will succeed in solving the converse problem of deducing (32), 

 (33) from (31). 



BoUzmann's Minimum Theorem. 



42. The property that the functional equation (30) is a necessary as 

 well as a sufficieiit condition of permanence was first proved byBoltzmann 

 for a single monatomic gas in 1872,^ for a mixture of two gases in 1886,^ 

 and both by Lorentz and by Boltzmann for a polyatomic gas in 1887.^ 



A similar investigation based on Boltzmann's was given by Bur- 

 bury in 1890. ' In his paper ' On the Collisions of Elastic Bodies,' 

 Burbury has adapted the proof to colliding systems in general, and a 

 similar generalisation is given in a better form by Watson.* Burbury's 

 specification of the states of the systems by generalised co-ordinates and 

 velocities, instead of momenta, is, to say the least, unfortunate, for the 

 complete investigation involves considerations not only of collisions, but 

 also of the free motions between collisions. Now, as we have seen in § 13, 

 the multiple difierential of the co-ordinates and m^omenta is an invariant in 

 such motions. But , the same is not necessai'ily true of the multiple 

 differential of the co-ordinates and velocities ; and even if the validity of 

 the argument in § 13 of Burbury's paper be admitted, it only applies to 

 rigid bodies under no forces. Watson obviates the difficulty by the use of 

 generalised momenta, and arrives at the following result. 



43. Let one of the co-ordinates q^ of one of the bodies be so chosen that 

 a collision occurs whenever q„ attains its maximum value zero. Let H 

 denote the function 



fF(log F-l)cZP, . . . ^Q,„.|- f/(log/-l) dp, . . . dq„ . (34) 



Then it is shown that 



dR 1 

 dt 



J-^(F'f-m log ^^^,dF, . . . dq„,dp, . . . dq„.,.q„ . (35) 



and the latter integral is essentially negative ; hence H diminishes with 

 collisions until F'f' — Ff=0. Also H is constant in the absence of colli- 

 sions, because the conditions of permanency then require F, y to be in- 

 dependent of the time. Moreover, the midtiple differentials c?P, . . . c?Q„ 

 and dp^ . . . dq„ are not affected by the choice of co-ordinates (§14 above), 

 and therefore no restriction is imposed on the generality of the conclusions 

 by choosing q„ to vanish for any particular collision under consideration, 

 nor does this choice affect the value of H. 



' 'Weitere Studien iiber das Warmegleichgewiclit unter Gasmolekiilen,' (SiYiJer. 

 der h. Wie7ier Akad., Ixvi. (ii.) (Oct. 1872), p. 275. 



- ' Ueber die zum theoretischen Beweise des Avogadro'schen Gesetzes erforder- 

 lichen Voraussetzungen,' Sitzher. xciv. (ii.), Oct. 188G. 



' 'Ueber das Gleichgewicht der lebendigen kraft unter Gasmolekiilen,' 'Neuer 

 Beweis zweier Siitze,' Sitzher. der k. Wiener Akad.,xcv. (iii.), ■fan. 1887, pp. 115, 153. 



* ' On some Problems in the Kinetic Theory of Gases,' Fhil. Mag., October 1890. 



' Kinetic Tlieory of Gases, p. 42. 



