254 PROFESSOR TAIT ON THE 



(B) That the particles of each kind, separately, acquire and maintain th.> 

 " special state." 



(C) That there is free access for collision between each pair of particles, 

 whether of the same or of different systems ; and that the number of particles 

 of one kind is not overwhelmingly greater than that of the other. 



Of these, (A) and (B), though enunciated separately, are regarded as conse- 

 quences of (C), which is thus my sole assumption for the proof of Clerk- 

 Maxwell's Theorem. Professor Boltzmann states that the only necessary 

 assumptions are : — that the particles of each kind be uniformly distributed in 

 space, that they behave on the average alike in respect of all directions, and 

 that (for any one particle X) the duration of an impact is short compared with 

 the interval between two impacts. His words are as follows : — " Die einzigen 

 Voraussetzungen sind, dass sowohl die Molekule erster als auch die zweiter 

 Gattung gleichformig im ganzen Raume vertheilt sind, sich durchschnittlich 

 nach alien Richtungen gleich verhalten und dass die Dauer eines Zusam- 

 menstosses kurz ist gegen die Zeit, welche zwischen zwei Zusammenstossen 

 vergeht. " 



He farther states that neither system need have internal impacts ; and that 

 Mr Burbury is correct in maintaining that a system of particles, which are 

 so small that they practically do not collide with one another, will ultimately 

 be thrown into the " special " state by the presence of a single particle with 

 which they can collide. 



Assuming the usual data as to the number of particles in a cubic inch of 

 air, and the number of collisions per particle per second, it is easy to show 

 (by the help of Laplace's remarkable expression * for the value of A "O m jn"' 

 when m and n are very large numbers) that somewhere about 40,000 years 

 must elapse before it would be so much as even betting that Mr Burbury's 

 single particle (taken to have twice the diameter of a particle of air) had 

 encountered, once at least, each of the 3.10 20 very minute particles in a single 

 cubic inch. He has not stated what is the average number of collisions neces- 

 sary for each of the minute particles, before it can be knocked into its destined 

 phase of the special state ; but it must be at least considerable. Hence, even 

 were the proposition true, aeons would be required to bring about the result. 

 As a result, it would be very interesting; but it would certainly be of no 

 importance to the kinetic theory of gases in its practical applications. 



I think it will be allowed that Professor Boltzmann's assumptions, which 

 (it is easy to see) practically beg the whole question, are themselves inadmissible 



* Thiorie Analytique dee Probability. Lime II, rha/i. ii, 4. [In using this formula, we must 

 make sure that the ratio m/n is sufficiently large to justify the approximation on which it is founded. 

 It is found to be so in the present case. At my request Professor Cayley has kindly investigated the 

 correct formula for the case in which m and n are of the same order of large quantities. His paper will 

 be found in Proc R. S. K, April 4, 1887.] 



