GASES ARISING FROM INEQUALITIES OF TEMPERATURE. 
241 
(4.) Mean Values of Combinations of £, rj, £. 
To find the mean value of any function of £, 77, £ for all the molecules in the element, 
we must multiply this function by f and integrate with respect to £, 77, and £. 
If the non-exponential factor of any term contains an odd power of any of the vari¬ 
ables, the corresponding part of the integral will vanish, but if it contains only even 
powers, each even power, such as 2 n, will introduce a factor 
R*0»(2w-l)(2w-3) . . 3-1 
into the corresponding part of the integral. 
First, let the function be 1, then 
or 
which gives the condition 
1 = 
d £ d rj d £ 
l = l+i(a=+/3'-+r) 
(23) 
(24) 
s 
°r + /3 3 +y 2 —0.. (25) 
Let us next find the mean value of £ in the same way, denoting the result by the 
ymbol £, 
£= (R0)’[a-L^(a 3 +a/3 3 -bay 3 )].(26) 
Since in what follows we shall denote the velocity-components of each molecule by 
w+£, v-frj, where u, v, w are the velocity-components of the centre of mass of 
all the molecules within the element, it follows that the mean values of £, 77, £ are each 
of them zero. We thus obtain the equations 
a+-|(a 3 -j-ay8 2 +ay 3 ) = 0 
/3+i(« 2 /3+/3 3 +/3y) = 0 l 
y+ U<*°Y+fiy+y s )=°. 
(27) 
Remembering these conditions, we find that the mean values of combinations of two, 
three, and four dimensions are of the forms 
P = R 0(1 + * 3 )1 
£77 =R 6<x/3 
e =(r ofod I 
frf=(B .0)**F l 
(qi = iXiSfafiy J 
(28) 
(29) 
MDCCCLXXTX. 
