OF THE KINETIC THEORY TO DENSE GASES. 
11 
x\ = x 1 — x (\x x + ny i + vzy) -\- x ( Xx . 2 + yy% + vz 2 ), 
y 'i — Vi — y (X x i + yyi 4" vz \) ffi y Q^ x i + yyz + vz z)> 
&c. 
That is, 
x\ = (1 — X 3 ) aq — Xyy x — Xvz v -f- X 2 x. 2 -j- X/xy 3 -f- Xcz 3 
y'i = — V^i + (i — y 2 ) y\ — y yz i + V^a + y°y-2 + y yz z 
z x = — Xvx 1 — yvy x + (1 — v 2 ) z x -4- Xvx 2 -f- y v y<z “h v ~ z z 
x o — Xhaq Xyy x -|- Xvz x -J- (1 — X-) cc 3 X/j.y .2 — Xzx 2 3 
y\ = + y 2 Vi + y yZ i — X i xx z + (1 — ff 3 ) y 3 — y yz % 
z % = Xvx x + nvy l + v\ — Xvx. 2 — [ivy 3 + (1 — J' 5 ) z 2 . 
Call these equations A. 
But inasmuch as the motion might take place the reverse way with. the same 
values of X, y, v, it must by the same reasoning be true that 
x x = (1 — X 3 ) x\ — Xyy\ — Xvz\ -f- X-x'. 2 + X/xy' 3 -j- Xyzh, &c. A', 
which are the same as equations A with the accents interchanged between the right 
and left-hand members. If we solve either set of equations we get the other set. 
17. Now it is given that at this instant the chance of the spheres having velocities 
x x . . . x x + dx 1} &c., is Ce -,lQ dx x dx z ... in which 
Q = ax 2 + bx x x 2 + ax. 2 2 -f ay x 3 + by x y 2 + ay 2 3 az x -j- bz x z z + az z 3 
-J- bx x (Xo -}- x 4 -}" &c.) -j- &c. 
-J- ax 2 + &c. + terms containing squares and products of the 
velocities x 3 . . . z n . 
The chance that after this collision the velocities of the n spheres shall have the 
values x\, y\, z' x , x' 2 , y\, z 2 , and x 3 , y 3 , &c., is found by substituting for x x , y x . z x , 
x 2 , y%, z 2 , in the index hQ, their values in terms of x\, y\, z\, x' z , y’ z , z' z , as given by 
equations A'. Effecting this substitution we find that the coefficient of x 2 is 
a . {(1 - X 3 ) 3 + X > 3 + XV + X 4 + X °> 3 + XV} 
+ b. {x 3 (i - x 3 ) - xy ~ xv] 
that is a. Similarly, we find that the coefficient of x\x z in the new index is b, and 
the coefficient of every product x x x r in the new index is the same as that of x x x r in 
the original index. The new index is then the same function of x\, y' x , z\, x z , y\ 2 , z\, 
that the original one was of x x - y x , z x> x 2 , y 2 , z z . The assumed distribution of velocit ies 
is therefore not affected by any one. and therefore not by any number of collisions. 
