392 Lord Kelvin [April 27, 



§ 50. The last case of partition of energy which we have worked 

 out statistically, relates to an impactual problem belonging partly 

 to Class I., § 28, and partly to Class II. It was designed as a 

 nearer approach to practical application in thermodynamics than 

 any of those hitherto described. It is, in fact, a one-dimensional 

 illustration of the kinetic theory of gases. Suppose a row of a vast 

 number of atoms, of equal masses, to be allowed freedom to move only 

 in a straight line between fixed bounding planes L and K. Let P 

 the atom next K be caged between it and a parallel plane C, at a 

 distance from it very small in comparison with the average of the 

 free paths of the other particles ; and let Q, the atom next to P, be 

 perfectly free to cross the cage-front C, without experiencing force 

 from it. Thus, while Q gets freely into the cage to strike P, 

 P cannot follow it out beyond the cage-front. The atoms being all 

 equal, every simple impact would produce merely an interchange of 

 velocities between the colliding atoms, and no new velocity could 

 be introduced, if the atoms were perfectly hard (§ 16 above), because 

 this implies that no three can be in collision at the same time. I do 

 not, however, limit the jjresent investigation to perfectly hard atoms. 

 But, to simplify our calculations, we shall suppose P and Q to be 

 infinitely hard. All the other atoms we shall suppose to have the 

 property defined in § 21 above. They may pass through one another 

 in a simple collision, and go asunder each with its previous velocity 

 unaltered, if the differential velocity be sufficiently great ; they 

 must recoil from one another with interchanged velocities if the 

 initial differential velocity was not great enough to cause them to 

 go through one another. Fresh velocities will generally be intro- 

 duced, by three atoms being in collision at the same time, so that even 

 if the velocities were all equal, to begin with, inequalities would 

 supervene in virtue of three or more atoms being in collision at the 

 same time ; whether the initial differential velocities be small enough 

 to result in two recoils, or whether one or both the mutual approaches 

 lead to a passage or passages through one another. Whether the 

 distribution of velocities, which must ultimately supervene, is or is 

 not according to the Maxwellian law, we need not decide in our 

 minds ; but, as a first example, I have supposed the whole multitude 

 to be given with velocities distributed among them according to that 

 law (which, if they were infinitely hard, they would keep for ever 

 after) ; and we shall further suppose equal average spacing in 

 different parts of the row, so that we need not be troubled with the 

 consideration of waves, as it were of sound, running to and fro along 

 the row. 



§ 51. For our present problem we require two lotteries, to find the 

 influential conditions at each instant, when Q enters P's cage — 

 lottery I. for the velocity (v~) of Q at impact ; lottery II. for the 

 phase of P's motion. For lottery I. (after trying 837 small squares 

 of paper with velocities written on them and mixed in a bowl, and 

 finding the plan unsatisfactory), we took nine stiff cards, numbered 



