48 
PROFESSOR G. H. DARWIN ON TFIE MECHANICAL CONDITIONS OF 
§ 15. On the Variation of Mean Frequency of Collision and Mean Free Path for the 
several sizes of halls. 
Each size of ball has its own mean frequency of collision and mean free path, and it 
is well to trace how the total means evaluated in the last section are made up. 
We have already seen in (48) that (substituting fur a its value in terms of 5 ) the 
number of collisions per unit time and volume between balls of sizes s to 5 + ds and 
balls of sizes s' to s' + ds' is 
IT* 
{x + yf (x^ + (xy}i e 
d.x dy, 
where x = sicr, y = s'jcr. 
But the number of balls of size 6' to 5 + ds in unit volume is 
Vi 0 _ 7 
_ X .-J/y% 
V TT 
Hence, the mean frequency of collision for a ball of size s with all others is 
Now, 
A.n 
(|7rF2) Y, 93 [ {x-\-yY {x^ + y^fx hjh yCly. 
Therefore, if we "write l/r for the frequency of collision of a ball of size s with all 
others, we have 
1 
T 
2h 
VJfff 
vilp 
[x + yY (F -|- y'^Y iT ^ 
Now, the mean frequency for all sizes is given by 
Hence, 
1 
T 
5-3318 
vcfl 
mip 
9w 
. TT" 
\ 8 /•OO 
% \ -nr r 
{x + yY {F + ifY 'IP dy. 
T _ 1 
T ~ 5-3318 3* 
= -l780-(^)f {x-{-yY{x^+ y^)Vj"e-yMy. . . 
(54) 
The integ”al involved here cannot in general be determined algebraically; but, if 
X be very small, or very great, we can find an approximate value for it. 
If X be very small, the integral becomes 
