30 
Proceedings of the Royal Society of Edinburgh. [Sess. 
These ratios can be readily calculated for various angles of incidence. 
If now we take the amplitude of the displacement in the incident wave to 
be unity, and rj 0 will be respectively the cosine and sine of the corre- 
sponding angle of incidence, and the corresponding values of the component 
surface displacements f and rj will follow at once. The values of these 
several displacements for different angles of incidence are given in the 
following table, together with the angle of emergence tan -1 (£/y ) : — - 
Angle of 
Incidence. 
Incident Displacements. 
Surface Displacements. 
Angle of 
Emergence 
tan -1 - . 
V 
CJ 
o' 
£o- 
Vo- 
I- 
V- 
0° 
1 
0 
2 
0 
o 
90 
QO 
10 
0-985 
0-174 
1-964 
0-398 
77-6 
4-935 
20 
0-94 
0-342 
1-86 
0-78 
67-2 
2-385 
30 
0-866 
0-5 
1-69 
1-122 
56-4 
1-506 
40 
0-766 
0-643 
1-476 
1-406 
45-5 
1-05 
50 
0-643 
0-766 
1-24 
1-616 
37-5 
0-767 
60 
0-5 
0-866 
1-0 
1-732 
30 
0-577 
70 
0-342 
0*94 
0-772 
1-71 
24-1 
0-451 
80 
0-174 
0-985 
0-528 
1-404 
20-7 
0-376 
90 
0 
1 
0 
0 
19-5 
0-354 
It is curious to note that, although both the component displacements 
of the surface vanish at grazing incidence, the ratio does not vanish. What 
this means is that the angle of emergence never becomes less than 19°'5, 
however large the angle of incidence may be. It is only at these approxi- 
mately grazing incidences that the angle of emergence differs markedly 
from the angle which the incident ray makes with the surface. A com- 
parison of the angles of emergence with the complements of the angles of 
incidence shows that they never differ by more than a few degrees so long 
as the angle of incidence is less than 70°. At incidence 60° the values are 
identical, the direction of the surface displacement being in line with the 
displacement in the incident ray. 
Thus it appears that the general conclusion to which I was led, although 
expressed not quite accurately in the former paper, is after all not far 
wrong. 
As established by the calculations now made, the conclusion may be thus 
expressed. When a plane condensational sinusoidal wave falls on the plane 
boundary of the elastic solid through which it is travelling, every point of 
the surface is thrown into a rectilinear sinusoidal motion, whose direction, 
for most incidences, makes, with the direction of the displacement in the 
incident ray, an angle not exceeding 4°'5, and generally much less. 
