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



kind is equal to the number which change from 0P 1 to — OP in 

 the reversed medium by collisions of the same kind, expressions 

 (2) and (3) are equal. Also u' in (3) is the same as u in (2) since 

 the velocity of the centre of mass is unchanged and the spheres 

 are elastic, and by ordinary geometry dS' is equal to dS and 

 d£dr)d£ . d%'dr)'d£' is equal to d^drj^^.d^dr/^d^. Hence, since 

 F(—0P^) in the reversed medium is equal to F(OP^) in the 

 original one, 



F(OP 1 )f(o Pl ) = F(OP)f(op), 



and therefore since the kinetic energy is unchanged by the impact 

 we have, by the ordinary methods, Maxwell's law of distribution 

 namely 



OP* 



F(OP) = Ae 



In order to examine the validity of the above proof the 

 assumptions underlying equations (1) and (2) should be further 

 considered. In equation (1) the assumption is that, if a number 

 of particles are distributed uniformly throughout a medium, 

 and if the velocities are so distributed that the number per 

 unit volume which have velocity op is f(op) d% 'drj 'dg ', then the 

 number in the volume F(0P)d^drjd^.7rs\(,dt, which may be 

 written do v is f (op) d^'drj'd^' . do v This assumption is equivalent 

 to two others, first, that the particles are distributed throughout 

 the medium with perfect uniformity so that we can safely take 

 the number of molecules of a given kind in an element of volume 

 to be proportional to the volume of the element if the element is 

 large enough to contain a great many molecules ; secondly, that 

 the particular volume do x is large enough for the first assumption 

 to be applied to it. In order to prove result (2) we must suppose 

 that the number of molecules within the volume 



j a 

 F(OP) d^drjd^. irs'udt . ^cos 6, 



which may be written do 2 , is proportional to do r Thus, since the 

 volume do t is greater than do 2 , the whole assumption that we 

 make is that do 2 , and consequently do 1 , is large enough to contain 

 a very large number of the molecules which are distributed so that 

 the number which have a velocity op is given by the function 



/(op)- 



To estimate the number of these molecules, suppose F(0P) 



N 0F * 

 to be equal to — — e * 2 , which is Maxwell's law of distribution. 



7r 2 a 3 

 Then, taking hydrogen as an example, since* Nits' 1 is 7"0 x 10 3 



* These numbers are calculated in accordance with Professor Tait's results, 

 Edin. Trans, xsxin. p. 91. 



