of Edinburgh, Session 1885-86. 399 
If v be greater than v v the value of this is 
-n^Sx^ l + |p); 
but if v be less than v v it is 
These values are equal, as they ought to be, for the case of v — 
We must now take account of the distribution of speeds among 
the particles in the layer. If there be n per unit volume, the num- 
ber having speeds between v x and v x + is 
1 *Jir . a 3 
Hence the logarithm of the fraction of the whole number of par- 
ticles, with speed v , which freely traverse the layer is 
4 mrs 2 
J 7 
-J X U' V ‘ ’"V' + A )^' +£*-**?■ T- + f>>)- 
If we write this, for a moment, as - eBx, it is clear that 
.-eSx 
6 
represents the fraction of a group of particles with speed v which 
penetrate unchecked the layer Bx, and thus 
s -ex 
represents the fraction which pass without collision through a dis- 
tance x. Hence the average depth to which particles with speed v 
can penetrate without collision is 
/ 
f ex dx 
e 
The value of e is, of course, a function of v. 
If we now suppose the impinging particles to have speeds assorted 
as they are in the statistically stable distribution, the average free 
