326 Mr Leahy, On the law of distribution of velocities [Mar. 7, 



number of molecules with velocity op struck by molecules moving 

 among them with relative velocity u is proportional to udt.nrs*, 

 secondly, when the distribution in the "reversed" medium was 

 taken to be the same as that in the original one. If the distri- 

 bution is not steady, expression (2) must be amended by inserting 

 a factor (1 + vdt), and expression (2) must contain a factor 

 (l + v'dt); where v, v depend upon the variations of F and f. 

 The result obtained as before will be that 



F(OP 1 )f(o Pl )-F(OP)f(op) 



will not be zero but equal to wdt where w depends upon v, v', F, 

 and f. Integrating equation (2) and using the proposition that 

 all directions of encounter are equally probable, we get the usual 

 result 



j t F(OP) = F(OP) d£d v dzf[Jd%'d V 'd?7rs 2 u 



i- \j\(F{OP\)f{o Pl ) - F(OP) f{op)) ds} , 



where dt~, drj', d£' are elementary increments of the components 

 of op, and the double integral is taken for all possible impacts 

 between particles whose velocities before collision were OP, op; 

 OP v op t being the velocities which the particles acquire if their 

 relative velocity falls within the cone of solid angle dS. 



Since the subject of integration in the double integral is equal 



d 

 to wdt, j,F(OP) must contain dt as a factor and will be very 



small when dt is very small. But, since there is a limit to the 



d 

 minimum value of dt, this does not prove -y-F{OP) to be zero; 



(Lb 



that is we cannot in this way prove the ultimate distribution to 

 be steady, although its variation from the steady state must be 

 small when the distribution of the particles is regular throughout 

 the space considered. 



Boltzmann's proof would show that the function H which he 

 has introduced will continually diminish until the steady state is 

 obtained, but I think that it assumes equations (1) and (2) to be 

 absolutely true, which they appear to be if the motion is from 

 the first assumed to be steady. The proof that the motion of the 

 particles finally must attain a steady state is apparently still 

 wanting, although the above argument shows that the divergence 

 from the steady state must ultimately be small. It is not im- 

 possible that F(OP) may ultimately be periodic with a period of 

 magnitude of the same order of magnitude as the time of free 

 path. But the assumption that the motion of the particles is 



