Illustration of the Theory of Gases. 437 



Assume 



/W <£.«-*<—*>*, (33) 



where <f>, ty are functions of t' only. Substituting, as before, 

 in (30), we find 



{i-W«-+) , }{i^+ 2 ^- 2 **} 



so that the equation is satisfied provided 



i~L +ip-2h*=0, (34) 



and 



^+2h+ = (35) 



The first is the same equation as we found before, and its 

 solution is given by (32) ; while (35) gives 



<yfr=xe- 2ht ' (36) 



Thus (32), (33), (36) constitute the complete solution of the 

 problem proposed, and show how the initial concentration at 

 u — a. passes gradually into the steady state when t' = co . In 

 the early stages of the process. 



^^T^'" 1 ^""" 1 .... (37) 



to which the factor N/Vtt may be applied, when it is desired 

 to represent that the whole number of masses is N. It 

 appears that during the whole process the law of distribution 

 is in a sense maintained, the only changes being in the value 

 of u round which the grouping takes place, and in the degree 

 of concentration about that value. 



There will now be no difficulty in framing the expression 

 applicable to an arbitrary initial distribution of velocity among 

 the masses. For this purpose we need only multiply (33) by 

 X(oi) doc, and integrate over the necessary range. Thus 



/( 



<f> being given, as usual, by (32). The limits for a are taken 

 + qo ; but we must not forget that the restriction upon the 

 magnitude of u requires that x( u ) sna ll be sensible only for 

 values of u small in comparison with v. 



