FOUNDATIONS OF THE KINETIC THEOEY OF GASES. 69 



/Jy? + y 2 + z 2 . But the argument above shows, further, that this density must 

 be expressible in the form 



/(»)/<*>/<■) 



whatever rectangular axes be chosen, passing through the origin. These joint 

 conditions give only two admissible results : viz., either 



f(x)=A, orf(x) = Be Cx *. 



The first is incompatible with the physical problem, as it would make the 

 percentage of the whole particles, which have one definite speed, increase 

 indefinitely with that speed. The same consideration shows a fortiori that, in 

 the second form of solution, ivliich is the only one left, C must be negative. 

 Hence the density of the distribution of " ends " already spoken of is 



If n be the whole number of particles, i.e., of "ends," we must obviously have 



/»" 



The value of the integral is 



A /!L- 



4VA 3 ' 



so that the number of spheres whose speed is between r and r + dr is 



lJMni-**i*dr (1) 



This distribution will hereafter be spoken of as the " special " state. 

 The mean speed is therefore 



2 



\lVs~ 



2 rhlr = - 



while the mean-square speed is 



"V IT J Z/l 



This shows the meaning of the constant h. [Several of the results we have 

 just arrived at find full confirmation in the investigations (regarding mixed 

 systems) which follow, if we only put in these P for Q passim : — i.e., pass back 

 from the case of a mixture of spheres of two different groups to that of a single 

 group.] 



4. Meanwhile, we can trace the general nature of the process by which the 

 "special" arrangement of speed expressed by (1) is brought about from any 

 initial distribution of speed, however irregular. For impacts on the containing 

 vessel do not alter r, but merely shift the particular " end " in question to a 



