46 THE DYHAMICAL THEORY OF GASES. 



velocities of a molecule should be arranged in this cycle, rather than in the 

 reverse order. If, therefore, the direct exchange between OA and OA' is not 

 equal, the equality cannot be preserved by exchange in a cycle. Hence the 

 direct exchange between OA and OA' is equal, and the distribution we have 

 determined is the only one possible. 



This final distribution of velocity is attained only when the molecules have 

 had a great number of encounters, but the great rapidity with which the 

 encounters succeed each other is such that in all motions and changes of 

 the gaseous system except the most violent, the form of the distribution of 

 velocity is only slightly changed. 



When the gas moves in mass, the velocities now determined are com- 

 pounded with the motion of translation of the gas. 



When the differential elements of the gas are changing their figure, being 

 compressed or extended along certain axes, the values of the mean square of 

 the velocity will be different in different directions. It is probable that the 

 form of the function will then be 



(27), 



where a, /8, y are slightly different. I have not, however, attempted to investi- 

 gate the exact distribution of velocities in this case, as the theory of motion 

 of gases does not require it. 



When one gas is diffusing through another, or when heat is being conducted 

 through a gas, the distribution of velocities will be different in the positive 

 and negative directions, instead of being symmetrical, as in the case we have 

 considered. The want of symmetry, however, may be treated as very small in 

 most actual cases. 



The principal conclusions which we may draw from this investigation are 

 as follows. Calling a the modulus of velocity, 



1st. The mean velocity is v = -/=a ..................... (28). 



Jir 



n 



2nd. The mean square of the velocity is v'^a 1 ....................... (29). 



2t 



3rd. The mean value of ( is f i = -a > ....................... (30). 



