92 PROFESSOR TAIT ON THE 



all of them are complex. The process of Boltzmann, alluded to in a foot-note 

 to the introduction (ante, p. 66), and which Clerk-Maxwell ultimately pre- 

 ferred to his own methods, involves a step of the following nature. 



An expression, analogous to the /of §3, but in which B and C are unde- 

 termined functions of the coordinates x, y, z, of a point, is formed for the 

 number of particles per unit volume, at that point, whose component speeds, 

 parallel to the axes, lie between given narrow limits. I do not at present 

 undertake to discuss the validity or the sufficient generality of the process by 

 which this expression is obtained, though the same process is (substantially) 

 adopted by Watson and others who have written on the subject. However 

 obtained, the expression is correct. It can be established at once by reasoning 

 such as that in §§ 2, 3, 4. To determine the forms of the aforesaid functions, 

 however, a most peculiar method is adopted by Boltzmann and Maxwell. 

 The number of the particles per unit volume at x, y, z whose corresponding 

 " ends " occupy unit volume at u, v, iv in the velocity space-diagram (§ 3), is 

 expressed in terms of these functions, and of u 2 + v 2 + w 2 . The variation of the 

 logarithm of this number of particles is then taken, on the assumption that 



Sx = u8t, &c, 8u— St, &c, 



ax 



where U is the external potential; and it is equated to zero, because the 

 number of particles is unchangeable. As this equation must hold good for all 

 values of u, v, w, it furnishes sufficient conditions for the determination of B 

 and C. The reasons for this remarkable procedure are not explained, but 

 they seem to be as below. The particles are, as it were, followed in thought 

 into the new positions which they would have reached, and the new speeds 

 they would have acquired, in the interval St, had no two of them collided or had 

 there been no others to collide with them. But this is not stated, much less 

 justified, and I cannot regard the argument (in the form in which it is given) 

 as other than an exceedingly dangerous one ; almost certain to mislead a 

 student. 



What seems to underlie the whole, though it is not enunciated, is a postu- 

 late of some such form as this : — 



When a system of colliding particles has reached its final state, we may assume 

 that (on the average) for every particle which enters, and undergoes collision in, a 

 thin layer, another goes out from the other side of the layer precisely as the first 

 would have done had it escaped collision. 



32. If we make this assumption, which will probably be allowed, it is not 

 difficult to obtain the results sought, without having recourse to a questionable 



