OF THE KINETIC THEORY TO DENSE GASES. 
9 
u x 
must be of the form 
u x -J- du x , &c., v x . . . v x -f- dv x , &c., 
Ce- 
h + b ]2 UiU 2 + o^u-z 2 + <fcc.) 
5 
in which the index is a quadratic function of u x , u. 2 , &c., v x , v. 2> &c., and iv x , w. 2 , See., 
but contains no products of the form uv, uio, or vw, and the coefficients b X2 , &c., are 
functions of the positions of the spheres. 
14. With regard to the forms of these coefficients, we observe that the quadratic 
function in the index must always be positive, because the chance cannot become 
infinite for infinite values of the variables, a. The condition for this is that the 
determinant 
D = 
2oq, b x2 , b x2 . . 
b X2 , 2a 2 , b 22 , 
and all its coaxial minors, must be positive, and, therefore, every a positive. 
15. Again b 12 , not being zero, expresses the fact that u x and u. 2 are not independent. 
But it is also a fact that they are more likely to be of the same than of opposite 
signs. Therefore, b 12 , and similarly every b, must be negative or zero. 
16. Evidently also the coefficients b must generally diminish in absolute magni¬ 
tude as the distance between the spheres to which they relate increases, and 
must become inappreciable at some distance, small compared with the dimensions of 
our system, but possibly large compared with the diameter of a sphere. The b’s 
must be functions of the positions of the n spheres within V having this property. 
Again we may consider either a volume V containing n spheres, or a smaller 
_ "j 
volume —— Y containing n — 1 out of the n spheres, that is, all the n spheres 
7b 
except one, which one belongs to the outer layer. If the velocities of that one 
be denoted by u,„ v„, to u , and if Q ;i be the quadratic function for the n spheres, Q«_! 
for the n — 1 spheres, we must have 
+ » 
e~" Q 't-i = | j | e~ h \ da,,, dv„, dw n . 
If we actually perform the integrations indicated for one variable, we get with 
in which 
Q ;i =: a x u x 2 + b x2 u x u. 2 -f- a. 2 uy + &c. 
Q«_i = cfpq 2 + b' x2 u x u 2 + a' 2 u 2 2 + &c., 
2 cc\ = 2a x — b ln z /2a„, 2ct 2 = 2a. 2 — b 2 „ z j2a„, 
b\ 2 = b 12 — b ]n b 2n /2a,„ &c. 
MDCCCXCYI.—A. C 
