50 
PROFESSOR 0. H. DARWIN ON THE MECHANICAL CONDITIONS OF 
These values are used for forming the curve, entitled “ frequency of collision,” in 
fig. 1 below, and they are supplemented by the values found above for J'/V, in the 
case where 5/5 is either very small or very larg’e. 
The frequency becomes infinite when the balls are infinitely small, because of the 
infinite velocity with which they move, and again infinite for infinitely large balls, 
because of their infinite size. But it must be remembered that there ate infinitely 
few balls of these two limitino’ sizes. 
We have now to consider the mean free path, say X, for the several sizes. 
If be the mean square of velocity for the size s, the mean velocity for that size is 
u y^(8/37r), by the ordinaiy kinetic theory. 
From the constancy of mean kinetic energy for all sizes, wm have 
so thah the mean velocity for size 5 is 
But, if U be the mean velocity, and L the mean free path, and T the mean free 
time for all sizes together, we have 
1'91.38 1'9138 T 
Therefore, the mean velocity for size s is 
W (s/Stt) 
1-9138 
Z 
T 
= -4815 
But the mean velocity for size s is X/V ; hence, 
-4815 1 ''''" 
L \s, 
2-705 
K 
4815 1 
1780 ■ K 
When s is very small, we find X/L = 4, and, when s is very large, X/L = 1-7 (yX)*- 
Thus, for small values of s, the mean free path reaches a constant limit 4, and for 
large values it becomes infinitely small. 
The intermediate values, sufficient for drawing a curve, are given in the following 
short table :— 
X/L. 
— Ln 
— 2? 
1-58. 
— 
— 4^ 
•93. 
= 9 
•55. 
— 4-0 
— 2^ 
•21. 
= 29 
•09. 
