in the Kinetic Theory of Gases. 89 



condition. The difference between the value which this quan- 

 tity has for a given condition and this minimum value gives 

 us a measure of how far this condition differs from the spe- 

 cial condition ; the value of the differential quotient of this 

 quantity by the time gives the rapidity with which this 

 condition approaches the special condition. 



To return again to the proof of Avogadro's law, in what was 

 said at the commencement, as well as in all my older papers, no 

 limiting assumption is made with regard to the ratio N x : N 2 , 

 but it is always taken for granted that the smaller of these 

 numbers is very large, though the greater may be many 

 times larger. Mr. Burbury* goes still further, since he 

 maintains that the two propositions given at the commence- 

 ment of this section also hold good even when of the one 

 kind of gas only a single molecule is present. With this 

 assertion, which of course has simply a mathematical interest 

 but no physical application, I have agreed, and do so still. 

 Since, however, the validity of this assertion is again an 

 entirely new question, on which the correctness of what has 

 been said does not in the least depend, I will not further 

 prolong the controversy by its discussion, and will only 

 remark that the objection of Prof. Tait with reference to the 

 reversal of direction of all the velocities, which cannot take 

 place until the occurrence of the special condition, and then 

 must affect not one only but all the molecules, has already 

 been answered in my paper, " Remarks on some Problems of 

 the Mechanical Theory of Heaf't- From an arbitrarily 

 chosen, not special, condition we pass under Burbury's 

 assumption (possibly not until after a very long time) into 

 the special condition. If, therefore, we were to reverse the 

 directions of all the velocities at the commencement of the 

 selected condition, we should, inversely perhaps, not reach (or 

 only during some time) conditions still further removed from 

 the special condition ; we should more probably, also in the 

 reverse direction, reach the special condition. 



§ 2. On the Proof of Maxwell's Law of the Distribution of 



Velocities. 



Prof. Tait admits, in his second paper, that his first proof 

 of this law is defective, inasmuch as the reason he gives for it, 

 that F(xyz) must be the product of three functions, of which 

 one contains only x, the second only y } and the third only z, 

 is not valid. By ~F{xyz)dxdy dz we are therefore to under- 



* Burbury, Phil. Mag. ser. 5, vol. xxi. p. 481 



t Boltzraann, Wien. Sitzber. vol. lxxv. sect. ii. (1877). 



