Probabilities to the Movement of Gas-Molecules. 265 



the disk w. Let the velocities after collision be W and w' — 

 say, W > w, w' > W 7 . Analogy with the simpler cases suggests 

 that we should equate 



F(U, v, a)f(u, c) = F(U', v, a,')f(u',v'). . (29) 



But there will not now be secured thereby the replenish- 

 ment of the class (U, V, ft) by means of collisions between 

 molecules of the dotted classes. The sort of collision which 

 might be expected to act thus is shown by reversing the 

 velocities of the points of contact, and then looking at 

 the figure from above (16). The view thus obtained is 

 shown in rig. 2 B, where the disk has the post-collision 

 velocities u 1 , v' ; but the velocity of 1' in the direction 

 perpendicular to A' B' is U' cos (f> + V'sin cf> minus eft, 

 where c is the perpendicular distance of the normal at 

 I from G, whereas in "W the sign of eft is plus. The 

 relative velocity of the points of contact at (i. e. just before) 

 this back-hand collision is thus not the same as it was for the 

 collision shown in fig. 2 A. The reasoning before employed* 

 thus breaks down. The break is obviated by the assumption 

 that the class (IT, V, — ft) occurs as frequently in the medley 

 as the class (U, V, + ft)'. We have then (30) F(U, V, ft) 

 = F(U, V, -ft). What is lost to class (U, V, ft) through 

 the collision shown by fig. 2 A is gained by the class 

 (U ', V, + ft')- And by the postulate the content of that 

 class keeps equal to that of (XT', V, —ft'). And collision 

 between molecules of the latter class and those of class 

 (u\v) — the collisions shown in fig. 2 B — results in the class 

 (U, V, —ft). If then equation (29) is satisfied, what is lost 

 to class (U, V, ft) through a back-hand collision will be 

 gained by class (U, V, -— ft)f. But the contents of these 

 two classes are equal by (30). Therefore if equation (29) is 

 satisfied, the class (IT, V, ft) — and likewise any other class — 

 will not be increased or diminished by collisions with m mole- 

 cules of the class (u, r) and likewise not by collisions with 

 those of any other class. 



The example may be generalized by supposing the ruler AB 

 to be connected by a hinge at B with another flat bar BC, and 

 that again with another link CD ; while the disk is likewise 

 connected with a complicated mechanism. By parity of 

 reason, the velocities lost to any class (Q„ Q 2 ...) through a 

 direct collision with any class (q\,</ 2 •••) — each class of an 



* Above, (22). 



f By analogy with the particular cases above or by the general 

 reasoning below (p. 267). 



Phil. Mag. S. 6. Vol. 40. No. 237. Sept. 1920. T 



