192 Proceedings of the Loyal Society of Edinburgh. [Sess. 
13. The full curve in the accompanying diagram represents the prob- 
ability of the emergent path being nearer to the centre than the entrant 
path ; the dotted curve represents the probability of the emergent path 
being further from the centre than the entrant path. Ordinates represent 
the probability, abscissae represent the angle 6. The probability is almost 
exactly 05 at 6 = 26°. 
If a and a are the complements of 6 and O', where O' is the smallest 
emergent angle corresponding to 6 as the entrant angle, we have a =3 a, 
provided that a is less than 7r/4. Conversely, no path a can be reached 
from a path outside the angle 3a. Thus, with all possible directions of 
entrant path, the probability that the emergent path shall have a small a 
is small. On the contrary, emergent paths for which a is large have large 
probability. 
Now the path for which the average radial and transverse components 
of kinetic energy are equal is given by 
tan 6 = '20 
and corresponds to 0 slightly larger than 23°. It therefore lies slightly 
inside the path ($ — 26°) which is equally readily reached from both smaller 
and larger values of 6. Therefore this latter path has a slight preponder- 
ance of the transverse component of kinetic energy over the radial 
component. The excess is scarcely one-fifth of the total kinetic energy. 
Consequently, all directions of incident path at the corrugated boundary 
being made initially equally probable, the large probability of emergent 
path with large a, that is, small 0 ; and the small probability of emergent 
paths with small a, that is, large 6 ; make it almost certain that the initially 
emergent paths will show an average preponderance of the radial over the 
transverse component of kinetic energy. This distribution of emergent 
paths constitutes a distribution of entrant paths still more favourable to 
that preponderance than was the initially postulated distribution. 
Without making a more detailed investigation, it therefore appears 
likely that the preponderance of the radial component, which was exhibited 
in Lord Kelvin’s tests, is a result to be expected. 
14. The corresponding result can be obtained more readily with a 
different law of reflection at the boundary. A molecule moving in the 
interior of a liquid or solid spherical cavity has probably no direction of 
reflection preponderant over another. We shall assume that all directions 
are equally probable for the reflected path whatever be the direction of the 
incident paths. If we assume, for simplicity, that the cavity has unit 
radius, and that the mass of the moving particle is 2 while its speed is 
