228 Proceedings of the Royal Society of Edinburgh. [Sess. 
indicates the value of the parameter which corresponds to each ray, the 
rays being represented by the terminal letter on the diagram : — 
Ray. 
A. 
B. 
C. 
D. 
E. 
Case I., parameter (p 
1*5 
1 
0-8 
0*5 
0-2 
Case II., parameter cp 
1-8 
1*7 
1-5 
1 
0-5 
To each full line OA, OB, OC, etc., there corresponds a dotted line Oa. 
Ob, Oc, etc., which starts tangential to the curved ray and is therefore the 
direction in which the disturbance begins to radiate outwards from the 
origin. Considerations of symmetry show at once that the angle which 
each dotted line makes with the surface at the origin is the same as the 
angle with which the ray emerges at the other end. In other words, 
this angle is equal to the emergence angle as tabulated above. 
In the diagram the left-hand semicircle shows the rays for the first 
case, in which the variation of speed is assumed to take place throughout 
the whole globe ; and the right-hand diagram the second case, in which the 
variation takes place only through the upper layer of thickness, equal to 
one-tenth of the radius. In the latter case the first ray OA lies wholly 
within the layer of varying speed of propagation, and is curved throughout 
its whole length. All the other rays represented pass partly through the 
interior of constant speed of propagation and are straight throughout a part 
of their course. Thus the rays OD, OE to distant points are very approxi- 
mately coincident with chords, but for shorter rays such as OC and OB the 
chordal coincidence is not so close. We shall discuss this case in some 
detail. 
The dotted line Oa in Case II. gives the direction in which a ray would 
have passed if the speed of propagation had been absolutely constant 
throughout the whole globe. This condition would have given rise to 
what we may call the spherical distribution of energy over the surface of 
the globe, half the energy being in fact distributed over the hemisphere of 
which the origin is the pole. But in the case represented in the right-hand 
semicircle the ray starting originally along the dotted line Oa becomes bent 
round by refraction so as to assume the position OA. The energy, of course, 
passes with it. Hence the energy, which in the simple case of spherical 
distribution would have been distributed over the part of the surface defined 
by the arc Oa with O as pole, is concentrated within the much smaller part 
of the surface defined by the arc OA. We are to imagine Oa to be one of 
a cone of rays which divides the spherical surface into two parts, defined 
