Probabilities to the Movement of Gas-Molecules. 253 



easily extended to the case of* radii and numbers which are 

 not the same, only of the same order of magnitude. 



Each set of disks is divided into classes defined by rates 

 of velocity. Thus one class consists of disks with velocities 

 respectively between U and U + AU, V and V + AV. The 

 distribution of these velocities is represented by the 

 frequency-function F(U, V) ; meaning that in a unit area 

 the number of disks of the class specified is NF(U, V)AUAV. 

 The function f(u, v) is similarly related to the other set. It 

 is required to determine the functions F and f so that the 

 distribution of velocities may be stable. 



Observing the unit area, note all the couples of disks 

 which are in such close proximity that within the short 

 time t (measured forward or backward from the present 

 instant) they either will come into collision, or have come 

 into collision. The species of couples so defined presents 

 two sections according as the partners are consilient or 

 dissilient, say positive and negative sections. 



Now let each species of couple be divided into varieties 

 (I. c. p. 260) defined by the mutual orientation of the disks 

 which are just coming into, or from, collision. Say a variety 

 consists of those couples for which the point of impact on 

 one of the pair is between s and s + As, s being the length 

 of an arc measured from a fixed point (rotation being ignored). 

 The content (the number of instances in a unit areaj of each 

 variety thus defined will not be the same; but whatever it is 

 it may be assumed to remain constant ; on the hypothesis of a 

 random distribution irrespective of position in space. Let 

 us designate the positive (consilient) section of a variety 

 (U, V ; u, v)s and the corresponding negative (u, v ; U, Y)s. 

 Let U', V and u', v' be the velocities which result from the 

 collision corresponding to the positive section of the specified 

 variety. Then the new velocities will belong to the section 

 (?/', v' ; U', V')s : the negative section of a variety which 

 may be tenned the reciprocal of the former. 



Observing the medley during the short time t, we may 

 expect that the whole initial content of the positive section 

 (U, V ; u, v)s will have passed into the negative section of 

 the reciprocal variety. Less than the whole content could 

 not have passed out of the specified section unless one of the 

 disks included therein were before collision wijth its partner 

 knocked away by a third disk. But this could only happen 

 through the very improbable double event of a third disk 

 being initially in the neighbourhood and also in such a 

 position and with such velocities as to hit one of the partners 

 within the time r. More than the whole initial content 



