Probabilities to the Movement of Gas-Molecules. 259 



and V their values in U, V, u, v above given are to be 

 substituted. It was thus shown that after many, say n, 

 collisions the velocity of the disk under consideration, say 

 U (,,) and likewise V ( ' f) , is a linear function of numerous 

 elements. 



It is now to be added that the plurality of constituent 

 elements is further secured in the more concrete case of 

 disks differing in type : say, one large class of mass M, 

 another of mass m. For then, after the first collision, 



U' = (M- m)(TJ cos 2 0+Y cos 6 sin 6) + U sin 2 



- Y sin 6 cos 0+2m(u cos 2 6 + v cos 6 sin 0), (19) 



with corresponding expression for V. Accordingly, except 

 in the limiting cases when M and m are very unequal or 

 nearly equal, the ultimate velocity of any disk is largely 

 made up of fragments of orders reduced, not only through 

 repeated multiplication by powers and products of sines and 

 cosines, but also by factors of the type (M — m) r , where 

 M— in is a proper fraction. The penultimate constituents — 

 the less remote progenitors — of the ultimate velocity do in- 

 deed bulk more largely. But in general, and except initially, 

 they are themselves the product of numerous antecedent col- 

 lisions, and are therefore on the way to normal distribution 1. 

 The system may be conceived as approaching that ultimate 

 state by successive steps as in the simpler case f, or rather 

 more rapidly. As before, we may argue that the normal law 

 in a general form will be set up ; and may determine the con- 

 stants from the conservation of energy, and the condition 

 that the velocity-distribution of the molecules which come 

 into collision during At should be the same before and after 

 collision. The frequency-function thus found may be written 



Const. exp.—iT/[T] (20) 



Here U, V and ?/, v are velocities relative to the centre of 

 gravity — which is at rest or moving; uniformly. The trans- 

 formation required when the velocities relate to a fixed point 

 has already been indicated J. 



II. This conclusion is reached more quickly by the second 

 argument. Putting /(U, V, v., v) for the sought function, 

 we have now to determine / so that the integral (over all 

 possible values of the variables) of j log/ should be a 

 minimum — subject to the condition that the co-extensive 



* Above, note to p. 251. T Above, p. 250. 



% Above, p. 253. 



