190 Proceedings of the Royal Society of Edinburgh. [Sess. 
The fact of the constancy of the result in twelve out of fourteen 
equal portions of 140 flights hears against Professor Magie’s supposition ; 
but, in view of the entire legitimacy of that supposition, it seems to be 
desirable to test the question farther mathematically. The following 
investigation indicates how Lord Kelvin’s result may be entirely 
correct. An extension on similar lines could fix the exact proportion of 
radial to transverse components of the kinetic energy. Without carry- 
ing out that extension, we obtain a disproof of the conclusion that all 
phases are equally distributed in space and time provided that all phases 
are possible. 
12. An incident path, making a fixed angle 0 with the line drawn from 
the centre of the circular area to the centre C of a semicircular corrugation, 
may cross the diameter A B of that corrugation at any point between A 
and B. All points are supposed to be equally probable. The diagram 
represents the case n = 5. The reflected path coincides with the incident 
path when it passes through C. As the crossing point passes down- 
wards from C, the reflected path makes a smaller angle, O', with 
C O until the point A 4 is reached, when the paths are contra-parallel. 
(Points such as A v etc., are indicated in the figure by their suffixes 
alone.) As the crossing point passes farther down, O' increases until 
reflection occurs at B. The magnitude of O' then diminishes until, 
when A 2 is reached, parallelism again occurs. Similarly A 3 and A 4 
correspond respectively to contra-parallelism and parallelism ; and so 
on. When the crossing point lies in the regions CA 1? A 2 A 3 , etc., 
the path which leaves the corrugation makes a smaller angle with 
C O than the entering path makes. When the point lies in the regions 
A 4 A 2 , etc., the emergent path makes a larger angle with C O than the 
entering path makes. 
In the same way we find that, when the crossing point is in the regions 
C B 1; B 2 B 3 , etc., the angle of emergence exceeds the angle of entrance ; while, 
when the crossing point lies in the regions B x B 2 , B 3 B 4 , etc., the angle of 
emergence is less than the angle of entrance. If the radius of the corru- 
gation be unity, twice the probability that the emergent path shall pass 
nearer the centre of the circular area having the corrugated boundary 
than the entering path passes is given by the sum of the lengths of the 
regions C A p A 2 A 3 , . . . . , B 4 B 2 , B 3 B 4 , . . . . The number of these regions 
depends on the value of the entrant angle, 0. When 0 has the value 7r/2 n, 
the points A 2n _ 3 , A 2(n _ 1) , B 2(n _ 1} , B 2n _ l5 coincide in pairs respectively with 
the extremities A, B, of the diameter. 
If p + 1 be the number of reflections of a path between its entrance 
