442 MR. S. CHAPMAN ON THE KINETIC THEORY OF A GAS 



Similarly, since U = V = W = 0, U' = V' = W = by definition, we have 



(24) 2a, + a,,, + a m -f a 133 = 0, 2a\ + a' m + a' 1!B + a' 1M = 0, 



and four similar equations.* 



We require the mean values of the following functions of U, V, W : 



IP = (2km)- 1 (1 + 0,0, UV = (2km)- 1 o u , U 3 = (2hm)- sl2 a m , UV 2 = (2hm)- 312 a l22 , 



U(IP+V 2 +W 2 ) = (2hm)- 3l2 (a m + a l22 +a lss ) = -2(2hm)- 3l2 a l ; 



similarly for the second system of molecules. 



As usual, 6 being the absolute temperature, we have (2h)~ l = R6>. The various 

 component^ of partial pressure due to the first gas, p ix , p zy , &c., are given by 



, p xy = 

 since p = vm ; the mean hydrostatic pressure p is given by 



P = %(p IZ +Pyy+P:*} = (3 + n 



similarly for the second set of molecules. 



In substituting^' in the equations (20), (21), we shall write 



(by equation 15) since FF', being the product of two first order small quantities, is 

 negligible. 



* It should be noted that the above expression for F is of the lowest degree consistent with the 

 satisfaction of the requirements. The function F must provide for small changes in the mean values of 

 even functions of U, V, W, and also of odd functions, both these changes being of the same order. The 

 terms of the second degree do this for the even functions, an, 22, ^33 being of the first order ; it might at 

 first sight be thought that the terms of the first degree would provide for the odd functions of U, V, W, 

 but this is not so on account of the conditions U = V = W = 0. Hence the terms of the third degree 

 must be present, and their coefficients must be of the same order as oj, a u , &c. Note added October, 1911. 

 See the note on p. 483. 



