398 
Proceedings of the Boyal Society 
Hence if there be a layer of thickness Sx , in which quiescent 
spheres are evenly distributed, at the rate of n x per unit volume, 
and if a group of spheres (whatever their common speed) impinge 
perpendicularly on the layer, the fraction of them which pass 
through the layer without collision is 
1 - n^rr^Sx , 
If they impinge obliquely on the layer, we must substitute for Sx 
the thickness of the layer in the direction of their motion. 
If the particles in the layer were all moving with a common 
velocity, we should have to substitute for Sx the thickness of the 
layer in the direction of the relative velocity. 
So far, all is so obvious as not to require proof. How suppose 
v to be the common speed of the impinging spheres, and that they 
all move perpendicularly to the layer. Also suppose that all spheres 
in the layer are moving with common speed v v but in directions 
uniformly distributed in space. 
Those of them which are moving in directions inclined from f3 to 
/3 + S/3 to the direction of motion of the impinging particles are, in 
number per unit volume, sin/? S/5/2 
The virtual thickness of the layer in the direction of relative 
motion is, so far as these are concerned, 
Sx ^ 1 ' 2 + V ^ ~ ^ VVl C0S ^ * 
v ’ 
the term involving the cosine having the negative sign, because the 
velocity v x has to be reversed in finding the relative velocity. 
Thus the fraction of the impinging particles which traverses this 
set without collision is 
-a \/ ?;2 + V ~ COS S . n . Q 
1 - nj rs 2 Sx — 1 sin /38p . 
All such expressions, from /? == 0 to /? = rr, each of them less than 
unity by an infinitesimal quantity, must be multiplied together to 
find the fraction of the impinging particles which traverse the layer 
without collision. The logarithm of this product is 
n-,rrs 2 Sxf — 
2vv 1 cos /? sin 0d/3 . 
