352 
Proceedings of the Royal Society of Edinburgh. [Sess. 
We have assumed, therefore, that the probability that the motion is 
along Ox is the same as the probability that the motion is along any other 
axis, and that all transverse velocities of either particle perpendicular to 
AB are equally probable. 
Let 
and 
x 2 + y 2 + z 2 = u 2 
r 2Q2 _j_ r 2 gi n 2 0^2 _ g2 . 
then, using square brackets to denote mean values, we have 
O 2 ] = |> 2 ] = [i 2 ] = J|> 2 ] 
and 
[r 2 # 2 ] = [r 2 sin 2 6<f> 2 ] = h\q 2 ] , 
where the mean values are formed according to the method laid down in 
Also we must clearly have 
L dt^ x \ 
— 0, etc., 
and the mean values of the six momenta and the fifteen products of 
momenta taken two together all obviously vanish. 
The assumptions made, when interpreted physically, represent the 
property that the molecule under consideration is assumed to belong to an 
isotropic substance. 
Let similar assumptions be made with regard to the force. 
The functions T > 1 (a) 1 , y v sq) and $ 2 (a3 2 , Vv z %) are constantly changing 
owing to the motions of the molecules, and it will be necessary to finally 
express our results in terms of the mean values of the differential coefficients 
of these functions with respect to the ’co-ordinates of the points. 
Let us assume that our external field of force is an equally probable one ; 
in other words, that the probability that the force has a certain value is the 
same as the probability that it has the reverse value, and that the probable 
force in one direction is equal to the probable force in another direction ; 
so that 
= 0, 
and since the axes might be taken in any directions, 
0<tq 
----- 0 and 
"a&f 
_0^ 1 _ 
dx 2 _ 
70^ 
_\daq 
<®iY] = r ( d hY 
J IKdzJ 
0 2 <£> 
also 
0 2 <t> 1 " 
_0aq 2 J L dj/] 2 
0 2< Jq 
0q 2 
0 2 <t> ] " 
3x^iy Y _ 
but 
0 2< fq 03> 2 
. 0aq ’ dx 2 _ 
4= 0 necessarily. 
