794 
PROFESSOR 0. REYNOLDS ON CERTAIN DIMENSIONAL 
necessary to obliterate the inequalities in the groups which pass through A in a unit 
of time, h will be infinite, and as I is so small that it may be considered as taking 
no part in the distribution, the rate of distribution will depend on the number of 
encounters in a unit of volume, and on some function/(a) of a, 
2a 
\/ vr 
being the mean 
velocity of the molecules. 
Therefore approximately 
dl 
dh 
-/to i 
(23) 
So that if T is the initial value of I, then after h encounters we have integrating 
and if h is infinite 
i=r e -/w 
1 = 0 
(24) 
f(a:) is a positive function of a, and is not a function of I; but both as regards 
form and coefficients /(a.) may depend on the nature of the quantity G. 
The question whether f(a) is different for any or all of the quantities n, v, v 3 &c., 
must depend on the nature of the action between the molecules during encounters. 
If therefore by comparing the mathematical results with those from experiments 
the several values of f(a) can be compared, a certain amount of light would be thrown 
on the action between the molecules. So far, however, the conditions of equilibrium 
in the interior of gas of which the temperature varies form the only instance in which 
the values of f(a) are brought into direct comparison. This instance affords means of 
comparing the values of f(a) for n and v 3 , and shows that these values must be equal. 
As regards f{a) for v or v 3 , there are no experimental results which furnish any 
further light than that f(a) has real positive values. 
These questions do not rise in this investigation, since f(a) for v 3 does not appear in 
the results, and should f(a) have a different value for v from that which it has for 
n and v 3 , the only result would be a numerical difference in certain coefficients as to 
the comparative value of which the experiment affords no approximate evidence. 
IV. That when the molecules which enter or leave an element of volume in a unit of 
time are considered separately, the proportion of the molecules (n v) entering in a unit 
of time in each entering group which will subsequently undergo encounters within the 
element, and the proportion of the molecules leaving in a unit of time, in each leaving 
group, which have undergone encounters within the element, are approximately pro¬ 
portional to the mean distance (Sr) through the element in direction of the group and to 
the number of molecules in each unit of volume of the element. 
V. That the mean effect of encounters in distributing the several inequalities of the 
molecules which, entering in a unit of time, encounter within the element is a function 
(/(a)) of the mean velocity of the molecules within the element at the instant, 
