352 FINAL STATE OF A SYSTEM OF MOLECULES IN MOTION 



Let a pair of molecules encounter each other, and after the encounter let 

 their component velocities be ', if,'. {,' and &', 17,', ,'. The nature of the 

 encounter U completely defined when we know - 17, -17,, ,-, the velocity 

 of the second molecule relative to the first before the encounter, and SG.-X,, 

 y,-y m , r, the position of the centre of the second molecule relative to the 

 first at the instant of the encounter. When these quantities are given, '-,', 

 i),'-i)' and {,' ,', the components of the relative velocity after the encounter, 

 are determinable. 



Hence putting a, ft. y for these relative velocities, and a, b, c for t Ir- 

 relative positions, we find for the number of molecules of the first kind having 

 velocities between the limits f, and , + </ &c., which encounter molecules of 

 the second kind having velocities between the limits and , + d, &c., in such 

 a way that the relative velocities lie between o and a + da, &c., and the relative 

 positions between a and a+da, &c. 



/, ( ? dfadl.f, (, 77, Q dtd-ndl.t (abcafr) dadbdcdadftdy... (4), 

 and after the encounter the velocity of 3f, will be between the limits f,' and 

 (\ + d(, Ac. and that of M t between the limits ' and ' + d, &c. 



The differences of the limits of velocity are equal for both kinds of mole- 

 cules, and both before and after the encounter. 



When the state of motion of the system is in its permanent condition, 

 as many pairs of molecules change their velocities from V u V t to F,', F,' as 

 from F,', F,' to F w V v and the circumstances of the encounter in the one case 

 are precisely similar to those in the second. Hence, omitting for the sake of 

 brevity the quantities d, &c., and <f>, which are of the same value in the two 

 cases, we find 



/, (& *, C,)/. (& * t) -/, ( vS, O/, (&', V. 4') (5), 



Anting log/(f, 17, = ^^^, I, m, n) (6), 



where I, m, n are the direction-cosines of the velocity, F, of the molecule M. 



Taking the logarithm of both sides of equation (5), 



F, (M, F.V.m.n.) + F t (M t F,*/,,*,) = F, (M, F^XV) + *\ (M n F t X) ( 7). 



The only necessary rektion between the variables before and after the 

 encounter is 



(8). 



