6 
and 
therefore 
MR. S. H. BURBURY ON THE APPLICATION 
cos 9 sin 9 cos 0 cl9 
m 2 
cos 9 sin 9 dO = — > 
o 
c cos (ul) 
Again whatever he the values of oj, xp, E, 
therefore 
cos (uq) = cos (it oj) cos ( a>< ]) 
= - cos (wo), 
co 
--- T 2c U --r 2c U ft) — -yfr cos E 
c cos (id) = —-cos (cor/) = —- 
v ’ 3 ft) x J ' 3 &) j 
We have to multiply this expression by the number of collisions which N spheres, 
each having absolute velocity oj, undergo in time clt with other spheres having- 
absolute velocity xp . . . xp + dxfj ; the angle between oj and xp being E . . . E -f dE, and 
then integrate according to xp and E. That number is 
N p7rc 2 q clt f(xp) dxji i sin E dE, 
if pf (xp) clxp be the number per unit of volume of spheres whose velocity is xp ... xp -(- dxp, 
r oo 
so that I f(ip) dxp = l. 
J 0 
Therefore the complete average value of c cos (ul), that is the average vertical 
displacement of the substituted spheres, is 
f 00 C tv nt 
clxp f (xp) sin E dE q — cos (ojq), 
o' Jo w 
or, since clt — ds/u, 
But 
&) — \p- cos E 
&) 
also 
so our result is 
f Npirc 3 dsfdxpf(xp) f ± sin E cl E 
Jo' Jo ® 
f i sin E dE q C0 ^ q) = f I sin E dE 
Jo OJ Jo 
f /WO dxp = 1, 
J 0 
N §7 rc 3 p els = kN ds. 
= 1, 
We know (Watson’s ‘ Kinetic Theory of Gases,’ 2nd edition, p. 56) that the 
quantity of vertical momentum transferred across the plane 5 = 0 per unit of area 
