82 REPORT— 1894. 



bowls, and the two-dimensional problem of circles in a plane having the 

 same property. Both these cases have been fully investigated by means 

 of long and complicated integrations by Boltzmann, ' who has also extended 

 his treatment to bodies of any shape, provided they are convex outwards 

 and have no sharp corners. 



The case of circular discs is also woi'ked out fully by Watson, ^ who, 

 liowever, takes the trouble of evaluating the functional determinant step 

 by step by expressing the final in terms of the initial velocities ; a process 

 which is obviated by the method of § 33 above. The frequency of col- 

 lisions of any kind is proportional to the relative velocity of the points of 

 contact in that kind of collision. It is found that the condition of 

 permanence will be satisfied if the number of discs whose three velocity 

 components lie within the multiple differential du dv do) is proportional to 



Ne-'''^ du. dv dio or N exp -\1tM.{n''- + v'^ + k^uj"^) . du dv doi . (27) 



whether the discs in collision are similar or belong to two different sets. 

 This is the Boltzmann- Maxwell Law. 



37. The case of lop-sided spheres formed the subject of a faulty 

 demonstration by Burnside, the result of which — quoted in my first 

 Report, § 44, third to twelfth line — was in contradiction to the Boltzmann- 

 Maxwell Law, and is now known to be incorrect. This has been show^i 

 by Watson,^ by Burbury,* and also in his aforementioned paper by Boltz- 

 mann. The correct result is that if the velocities and anjrular velocities 

 about the principal axes be arranged according to the Boltzmann- 

 Maxwell distribution 



l^e'^'"^ dudvdiv dio^dw^db)^ . . , , (28) 

 that is 

 'Nexp — lh{M.(^t,■ + v'^ + w''-) + Alo^'^ + Bu^o^ + C(i);i'^} . du dv dw djLi^ dw^dui^ 



this distribution will be unaffected by collisions. From Appendix A we see 

 that it will also be unaffected by the free motion of the spheres between 

 collisions, and therefore it satisfies all the necessary conditions of perma- 

 nence. The mean values of Mu"^, Mv-, Mw-, Aw,-, Bw2^, Cwj^, are 

 equal. The other distribution of Appendix A, in which the mean kinetic 

 energies due to the three principal rotations are unequal, is, in general, no 

 longer permanent when collisions take place. In fact, Boltzmann starts 

 by assuming a distribution of tlie form 



:i^i!J!uAiJ^exp-hy + v'^ + w'-\-kiw^^--k^u,^^-k:iW./-.du. . . dw^ 

 and deduces that 



7 /C J ft n rCi 



^^=i=B=c- 



Now evidently this conclusion does not necessarily hold good if each 

 molecule is symmetrical about the line joining its centres of inertia and of 

 figure, because the angular velocity about this line will be unaffected both 



' ' Ueber das Gleichgewicht der lebendigen Kraft zwischen progressive! unci 

 Rotations-Bewegung bei Gasmolekiilen,' Sitz. d. k. Akad. zu Berlin, Dec. 1888. 

 '■* Kinetic Theory of Gases, p. 15. 

 » Natnre, vol. xlv. March .31, 1892, p. 512. 

 * Ibid., vol. xlv. April 7. 1892, p. 533. 



