98 BEPURT— 1894. 



condition that if the kinetic energy be reduced to a sum of n squares, 

 the mean values of these squares are equal. Hence the conclusion may 

 be stated thus : — 



If the necessity for making 8Q 'T a complete differential can be 

 established as a substantive law by independent evidence, the investiga- 

 tion affords an independent proof of Maxwell's law of j)artition of kinetic 

 energy between the momentoids for such a system. 



Conclusion. 



60. The conclusions arrived at in the present Report are to be regarded 

 as superseding the statement of Clerk Maxwell's Theorem in §40 and the 

 greater part of §§ 41, 42, 4 3 of Section III. of my first Report. The rest 

 of that Report is not, so far as it goes, materially affected by any results 

 now established, although several important questions connected with the 

 Boltzmann-Maxwell Law have now received a definite answer. 



The proof of the law and the assumptions involved in it are fairly satis- 

 factory for gases whose molecules collide with each other to a certain 

 extent at random, but in a medium in which the molecules never escape 

 from each other's influence the subject still presents very great difficulties. 



Even should it be shown that the law cannot be disproved for such a 

 medium, there still remains the question as to whether the distribution is 

 the unique one satisfying the conditions of permanence. The general 

 question of uniqueness, even in souie of the cases where the law admits of 

 more or less satisfactory proof, still suggests some questions for investiga- 

 tion. Intimately connected with this is the difficult question of stability. 

 For example, when a gas is condensed, its density at any point at first 

 remains proportional to e~'"- in accordance with the Boltzmann-Maxwell 

 Law ; but when a certain stage is reached, instability sets in, and part 

 of the gas liquefies. If the Second Law be true, the new distribution 

 satisfies Maxwell's law of partition of energy. Does it likewise satisfy 

 the Boltzmann-Maxwell Law ? 



The connection with the Theory of Probability still suggests subjects 

 for research. The relations of electrical and optical phenomena to the 

 Kinetic Theory open up an almost unexplored field. 



61. It only remains for me to thank all those who have assisted me in 

 collecting materials for this Report. I am particularly indebted to 

 Dr. Ludwig Boltzmann for his kindness in sending me copies of nearly all 

 his writings, and for several valuable suggestions that have helped to 

 clear up difficulties in the work. My thanks are also due to Mr. Burbury, 

 Dr. Ladislaus Natanson, Professor Sydney Young, and others for similar 

 help, which has very materially lightened the work of consulting and 

 examining the large mass of existing literature relating to this most 

 interesting branch of Mathematical Physics. 



APPENDIX A. 



Th'i Possible Laics of Partition of Rotatory Energy in Non-colliding 



Rigid Bodies. 



The motion of a rigid body about a fixed point or about its centre of 

 mass under no forces affords one of the best test cases bearing on Maxwell's 

 law of partition of kinetic energy. 



