THE DYNAMICAL THEORY OF GASES. 47 



4th. The mean value of f 4 is ^ 4 = |a 4 ................................. (31). 



5th. The mean value of y is |y = i a 4 ................................. (32). 



6th. When there are two systems of molecules 



Mjaf =M& .................................. (33), 



whence Mj)? = M t v; .................................. (34), 



or the mean vis viva of a molecule will be the same in each system. This 

 is a very important result in the theory of gases, and it is independent of the 

 nature of the action between the molecules, as are all the other results relating 

 to the final distribution of velocities. We shall find that it leads to the law 

 of gases known as that of Equivalent Volumes. 



Variation of Functions of the Velocity due to encounters between the Molecules. 



80 

 We may now proceed to write down the values of -~ in the different 



cases. We shall indicate the mean value of any quantity for all the molecules 

 of one kind by placing a bar over the symbol which represents that quantity 

 for any particular molecule, but in expressions where all such quantities are to 

 be taken at their mean values, we shall, for convenience, omit the bar. We 

 shall use the symbols 8 1 and 8., to indicate the effect produced by molecules of 

 the first kind and second kind respectively, and S 3 to indicate the effect of 

 external forces. We shall also confine ourselves to the case in which = 5, 

 since it is not only free from mathematical difficulty, but is the only case which 

 is consistent with the laws of viscosity of gases. 



In this case V disappears, and we have for the effect of the second system 

 on the first, 



where the functions of rj, in j(O/-Q)d<f> must be put equal to their mean 

 values for all the molecules, and A 1 or A^ must be put for A according as 

 sin* Q or sin" 20 occurs in the expressions in equations (4), (5), (6), (7). We 

 thus obtain 



w "'-'*^'*- 6 ' ................................. (36); 



