108 Distribution of Velocities in a Molecular Chaos. 



More generally, if Z, rn, n ... being any r (finite, not very un- 

 equal) factors with sum. of squares equal to r, are sporadically 

 attached to different elements in the successive batches, this 

 modification of the elements will not sensibly alter the law 

 of frequency for the compound. If the system of elements 

 thus modified is similarly sprinkled with another, and another, 

 set of similar factors X, fi, v . . ., the character of the compound 

 still persists. 



To apply this theory, let us begin by supposing the molecules 

 to be perfectly elastic equal spheres. And let us simplify 

 enunciation by conceiving a molecular chaos in two dimen- 

 sions — as it were an enormous number of perfectly elastic 

 billiard-balls repeatedly colliding with each other as they 

 dash about at random over the smooth surface of an immense 

 billiard-table with perfectly elastic cushions (compare Jeans, 

 ' Dynamical Theory of Gases,' p. 4). The centre of gravity 

 of the system may be supposed to be at rest. Let it be 

 granted {a) (as usual, in accordance with the fact that the 

 pressure of a gas at rest is equal in all directions) that the 

 mean square of velocities resolved along each of two rect- 

 angular axes, OX and OY, is the same, (b) that there is 

 reached a stable distribution of velocities, the same for 

 velocities in either direction. Then, if possible, let that 

 final law of frequency be other than the Gaussian law. At 

 any time, T, after the state of complete chaos has been 

 reached, consider the collision of two molecules of which 

 the velocities are respectively (resolved in the direction of 

 the two axes) n s , v s and u t , vt; quantities which may be 

 regarded as taken at random from the melange to which the 

 final law of frequency is supposed to apply. The velocities 

 after collision will depend not only on the velocities before 

 collision, but also on the direction of the line joining the 

 centres of the spheres at the moment of contact. Say that 

 line makes the angle 6 with the axis OX. Then the velocities 

 of the ball, which before the collision were u s v s , become 

 after collision 



w/= (u t cos + vt sin 0) cos 6 + (u s sin 6 — v s cos 6) sin 6, 

 vj = (u t cos 6 + vt sin 6) sin — (u s sin 6 — v s cos 0) cos 6. 



Likewise the expressions for u t ' v t f are linear functions of 

 the four velocities before the collision. Also other pairs of 

 colliding particles acquire similarly compounded velocities. 

 Thus after numerous collisions the velocity of a molecule in 

 either direction may be regarded as a linear function of 

 elements distributed according to a definite law of frequency, 



