Probabilities to the Movement of Gas- Molecules. 245 



Thus the frequency-function is of the form (8) Const, exp 

 — XT ; where T is the kinetic energy (7), and X is deter- 

 mined by the condition that the energy of the system for 

 any assigned values of the co-ordinates is constant (I. c. 

 p. 263). 



This reasoning may at once be extended to encounters in 

 the case of motion in one dimension since the relevant 

 equations (3) are in that case the same for collisions and 

 encounters. In the case of disks on a plane the conservation 

 of energy and that of .momentum give only three equations 

 for the four resultant velocities U', Y\ u\ v . A fourth 

 condition is presented by assigning the angle 26 through 

 which the direction of the relative velocity is turned by a 

 collision or encounter. There can therefore be always 

 arranged a fictitious collision which has the same effect in 

 changing the velocities as any given encounter. Replacing 

 each encounter by the equivalent collision, we can transfer 

 to encounters the conclusion which has just been proved for 

 collisions. An equivalent collision is not so simply obtained 

 when there are several degrees ol freedom. Let us pass to 

 this case by supposing one set of disks to have its centre 

 of mass different from the centre of shape. The angle 

 between the line joining those points and a fixed axis, say </>, 

 defines the phase of an encounter. We may now conceive 

 a second disk of the other (symmetrical) type to be in con- 

 tact with the first at any point on the periphery thereof. 

 But we thus obtain only four equations to determine the 

 five quantities XT', u', V, v and <£' the changed velocity of 

 rotatiou. To obtain another equation let us artificially 

 create a second point of impact by affixing a sort of buffer 

 or tentacle (of negligible mass) to the first disk and arranging 

 that there shall be simultaneous impacts at two points. 

 Generalizing the formulas given for impact at one point 

 (I. c. p. 262), we shall now obtain 11/, 11/... 7r/ in terms of 

 IIj, II s ..'.7ri... and two sets of the coefficients -called A, X, 

 each set depending on the position of a point of contact. 

 We have thus with the three conditions given by the con- 

 servation of energy and that of momentum two additional 

 variables, whereby to secure an impact equivalent to the 

 encounter. It would .not, however, be an exact equivalent. 

 For the angle cf> would in general be changed by nn en- 

 counter, but not by the impact. But since the values of </> 

 occur with uniform distribution, the frequency-distribution 

 of the actual system would be the same as that of the 

 artificial system. The reasoning may be extended to the 

 general case where there are several " momentoids." 



