242 Prof. F. Y. Edgeworth on the Application of 



This may mean that, if we select all the members of one 

 class (in a unit of the medley), say those between U and 

 U-S- AU, and observe the velocities of those molecules belong- 

 ing to a different set which are nearest to a member of the 

 selected class, the distribution of the array thus presented 

 will not be that of the i^-velocities in general, but one more 

 closely congregated about a centre, rU. 



The essential condition for this consummation is the linear 

 relation between the velocities U and u of two particles about 

 to collide and the resulting velocities U' and u' : 



U / = (M-m)U + 2mwl 



M ' = 2MU + (m-M)J ; ....() 



il M + m = l {loc. cit. pp. 250, 251). 



When we go on to two dimensions, considering first 

 perfectly elastic circular disks moving over a perfectly 

 smooth plane/ the essential condition still persists. But the 

 coefficients of the linear function are not now as in the 

 simplest case (3) the same for every collision. They involve 

 a variable parameter, the sine or cosine of an angle #, made 

 by the line of centres at the moment of collision with the 

 direction of the relative velocity (or with a fixed axis) (I. c. 

 p. 259). But the presence in the coefficients of such a 

 parameter hovering in random fashion about an average is 

 not fatal to the genesis of the law of error. This was 

 pointed out in the original paper (p. 262) with reference to 

 a later stage, the impact between molecules with several 

 degrees of freedom. The principle should have been intro- 

 duced at an earlier stage ; and it might have been employed 

 at a still later stage, namely with reference to encounters as 

 distinguished from collisions. 



Consider molecules with several degrees of freedom such 

 as chains — bars linked by hinges — moving over a perfectly 

 smooth plane. Let the generalized components of momentum 

 Pi, P 2 ... P 5 ; Pi, P2-->Pd De changed by an impact to 

 P/, P 2 '...; Pi, P2 1 ■■•• Each of the latter set may be' 

 regarded as a linear function of the original components ; 

 the coefficients now depending not only (as in the simple 

 case of disks) on the relative position of the mass-centres 

 (before the encounter), but also on the co-ordinates. Still 

 the coefficients of the linear functions connecting the P"s 

 with the P's may be regarded as hovering about a mean value ; 

 and accordingly the law of error will be set up. This 

 reasoning reinforces the proof which was before offered in 

 the case of encounters (I. c. p. 268); based on the mean 



