92 BEPOKT 1891. 



dimensions, the molecalar coordinates (.r, y, z) mnst fluctuate between 

 certain finite limits, and hence 8a;, 8y, Sz, cannot increase indefinitely with 

 the time. Hence by taking the time i sufficiently large we must have 

 ultimately 



^ Vm(rcSa! + i/S(/ + 3Sz) 



£..} 1 — ^=« ■ • (i^> 



since the numerator does not increase indefinitely with i. 



Now, it appears to me that the statements printed in italics are open 

 to objection. There is no reason why 8,v, 8y, Sz should not increase con- 

 tinually with the time until they can no longer be regarded as small' 

 variations, and it seems highly probable that this ivill happen under 

 certain circumstances. Take, for example, the case of a gas formed of a 

 number of hard spherical molecules colliding with one another, the 

 lengths of the mean free paths being great compared with the radius of 

 each sphere. If the direction of motion of one of these spheres be varied 

 very slightly, then at the next impact there will be a considerable altera- 

 tion in the direction of the line of centres.* After the impact, therefore, 

 the variation in the direction of motion will be very greatly increased, 

 and a similar increase will take place at each impact, until at last the 

 molecule will no longer collide with the same molecules as in the original 

 motion, but will come into collision with quite a different set. By this 

 time there will not be the slightest connection between the original and 

 the varied motion. 



15. I would therefore suggest that the existence of a ' quasi-period ' 

 i, as defined by (13), can be better explained by arguments of a statistical 

 nature based on the immensely large number of the molecules present in 

 a body of finite dimensions. In the steady or stationary motion of such 

 a body, it is reasonable to assume (as in the kinetic theory of gases) that 

 the velocities of the molecules are on the whole equably distributed as 

 regards direction. Thus, for example, the average number of molecules 

 for which x is positive and lies between ?t and u-^du is equal to the 

 average number for which x is negative and lies between —u and 

 — {u + dii) . 



Moreover, in the disturbed motion the displacements (Ss, hj, 82) of 

 any molecule cannot depend in any manner on its velocity components 

 (x, y, z). It is of course quite possible to conceive a disturbance of the 

 motion in which some fixed relation exists between the displacements and 

 the velocity components of the molecules — indeed, we might choose the 

 relation to be any we please — but a disturbance of this kind would only 

 be possible if the molecules were individually controllable ; in other words, 

 the displacements could only be brought about by means of Clerk Max- 

 well's ' demons,' and it would then be reasonable to suppose that the 

 Second Law would fail altogether. 



Hence in any physically possible variation of the motion the terms 

 involving positive and negative velocity components in the expression 



'^miiSx + yhj H- z8z) 



■will on the whole cancel one another, and therefore the average value of 

 the expression will be zero. This proves Clausius' Theorem. 



' This is easily exemplified by means of billiard-balls. 



