Distribution e~ 2h * and van der Waals' Theorem, 413 



of Q, namely, 



Q = Sra (u 2 + v 2 + w 2 ) 4 ^%b ik (uiU k + ty-j + tt^) , 



where h k is a function of the distance r t k between the two 

 molecules to which i and k relate. 



I here reproduce the proof given in Chapter VIII. of my 

 Treatise on the Kinetic Theory of Gases. Let D be the 

 determinant of the coefficients in Q, that is, let 



2m, b 12 b n ... 

 D = b l2 2m 2 b n . . . 



b n b 23 2m 3 . . . 



and let D n D 12 &c. be its first minors. 



For any values of the coordinates, we must have, as in 

 Boltzmann's theorem, 



A If tr^dux . . . dw n -l. 



In order that this may be true with my value of Q, A 

 must contain \/D as a factor. I will therefore write A \/D 

 instead of A. Then \ f D and Q are functions of the coordi- 

 nates only as contained in the b's, that is only as contained in 



doc dti 



the r's. As before, for each molecule , =u, and m-j- 



= j^ &c. But in the present case -~^ is not, as in 



ax du x 



Boltzmann's case, 2mu lf but -: — =2mu 1 + b 12 u 2 + b ls ii 3 + &c. 



(In i 



We find, however, taking mean values, that given u u 



— = — ie l9 where D n is the first coaxial minor of D. 

 etui J-'n 



The proof now differs from Boltzmann's in that we have to 



differentiate according to x, not only A but also Q and v'D. 



But since Q and \/D are functions of the coordinates, only 



as these are contained in the r's, the terms derived from 



~ and — i — can, as in Art. 4, be reduced to pairs each 

 ax ax 



containing a factor of the form -2-£ — , and therefore 



Vpq 



separately vanish if the motion be stationary. We have 

 then to make 



x \ ax "'l-L'n dx ) 



