1 
46 
PROFESSOR G. H. DARWIN ON THE MECHANICAL CONDITIONS OF 
It follows, therefore, that in our case collisions are more frequent than if the balls 
were all of the same size in about the proportion of 4 to 3. 
In order to find the mean free path, we require to find the mean velocity. 
If be the mean square of the velocity for any size s, the proportion of aU the 
spheres of that size which move with velocities lying between v and v + dv is 
“/“ dy, 
\/ TT 
where = 3'r^/2w®. 
But the number of spheres of size between s and s -j- ds, in unit volume, is 
O 7 
x-'e ^ dx, 
y/tt 
wBere x = sja. 
Hence, the mean velocity U is given by 
j" I vx^y''''e~^~y'' dx dy. 
U 
Now, 
so that 
Therefore, 
But 
therefore, 
V = ^/f . uy, and = 9 ^ V^, or x^u^ = (-) V^, 
2 3 2/2 
u = ^ x~^^ V, and v £r^ x~hy V. 
TT 
7rV3 
U=''^V\ f x^fiT^-fdxdy. 
Jo Jo 
yh f dy = i and f 
J r 
xh dx= 2 z~e~-" dz ; 
U = 
32^2 
tt'y/S 
^- v\ zh-^dz. 
3 J n 
This integral may be evaluated as follows 
Let 
J=[ x^e ^\lx, 
pQO ^00 
4=4 x^y~e~-^'~^ dx dy 
J n J 0 
?‘^sin^ 29 -4sm2 2fl) 
= 17 " 
•'0 Jo 
— 1 
— 4 
0 •'0 
JqJo (1-isii/0)’ ^ 
, ish9cf) 
Jn (1 - isin“(t)’ ^ 
= 4-4 
- A4T 
f’-_#_ p 
J A Q — i sin^ 4 ))^ 
L*' 0 
