538 
Proceedings of the Royal Society 
exponentials. Then, for a v and a v v colliding, the mean value of 
u 2 , the square of the speed of the v in the line of centres, is 
fnnf sin fid/3 . v 2 cos 2 a . sin adadcf> 
f nnf sin /3d/3 sin adad<f> 
Here the limits of a ought to he restricted to a hemisphere of 
which the direction of relative motion is the axis. But, by extend- 
ing them to the whole sphere, we merely double each integral, while 
we avoid some complex relations. We may compensate for this by 
taking <£ from 0 to tt only. [The same remark applies to the other 
integrals of this kind with which we have presently to deal.] The 
above fraction then takes the value 
Jnnf sm/3d/3. 
Since v 2 = v 2 + v\- 2vv 1 cos ^ , 
this becomes 
In further integrating this, we must remember that the limits of v 
are v - v x and v + while v > v l , but v 1 — v and v 1 + v while v< v 1 . 
Thus the denominator (being symmetrical) consists of two parts, 
which differ from one another only by the interchange of p and q. 
The value of the numerator can be found from that of the denomi- 
nator by differentiation with regard to p and change of sign. When 
this is done we have for the mean value of u 2 
3 p 4 - 4 q 
6p{p + q) 
This shows how the first and, by parity of reasoning, the second 
terms of (1) are affected by the new consideration. We might have 
directly verified this for the second term by evaluating 
f nnf sin /3d/3 . v{(cos a cos /3 -I- sin a sin /3 cos <j>) 2 . sin adad<f > , 
J'nnf sin /3d/3 sin adadcf> 
