44 
PROFESSOR G. H. DARWIN ON THE MECHANICAL CONDITIONS OF 
But 
o vr- o , 64 TT- 32 2^^ 
= -W ^ ’ and {t^y -j = Za • W 
TT* 
TT* 
Hence, the number of collisions per unit time and volume between spheres whose 
radii range between and + cls^, and others with radii between Sg and + dso, is 
^ ^ + yy + y^y {^yy e~'^~^\lxdij. 
The number of collisions of a single sphere per unit time is ijn of this, and, since 
n — p/m, we have for the collisions of a siimle sphere the factor instead of 
O J. ^J^jp 
Then the total number of collisions of all kinds in unit time, or the reci23rocal of the 
mean free time, is the double integral of this from oo to 0. 
For the ]3ur]30se of carrying out the integration, we may conveniently, as an 
algebraic artifice, change from the rectangular axes x, y to the polar coordinates r, 6. 
Thus, 
«CO ^00 
(a; + yy (x? + y^y [xyy dx dy 
0 ■ '* CO J 
— [ g-'-" dr [ (sin 6 + cos Oy (1 — sin 9 cos Oy (sin 6 cos Oy dd. 
Jo Jo 
Now, if we put r = z^, 
9.5.1 
f r — 2 [ e ^ dz — 2 • I e ^ dz. 
Jo Jo 4.4.4J0 
For the transformation of the second integral, ]Dut 
2 ; = cos 9 — sin 9, 
and we find 
[ (sin 9 + cos 9y (1 — sin 9 cos d)- (sin 9 cos 9)' d9 = \ 2 (2 ~ (1 “ ^^y dz 
Jo J -1 
= [ (2 — z^y (1 — z^y dz. 
Jo 
Hence, the whole integral is 
II C e-^ dz f (2 - z^y (1 - zj dz, 
Jo Jo 
and the mean frequency of collision of a single ball ^^er unit time is 
[ 6“"^ dz [ (2 — 2 :^)^ (1 — 2 “^)^ dz. 
J n J 0 ^ 
15. 
ttA m/p 
