222 Proceedings of the .Royal Society of Edinburgh. [Sess. 
Finally, we have by equation (4') for the angle of emergence 
e = cos -1< ff = cos -1 $ x /l*2 = cos -1 (l’095<£) . . (10) 
/X 
From these four equations (7), (8), (9), and (10) we can quickly calculate 
for each assumed value of <p between zero and 2'236 the important character- 
istics of various rays. The results are given in the following table : — 
Case I. v = 13’6 J 12 — x 2 throughout the globe. 
Parameter 
0. 
Arc 
2 0°. 
Transit time 
2T min. 
Minimum 
radius 
X 1‘ 
Emergence 
angle 
e. 
Energy distribution 
over surface 
defined by arc. 
2*2 
5-03 
1-6 
0-998 
10-3 
326 
2 
io-i 
2-9 
•98 
26-6 
20-04 
1*8 
15-6 
4 3 
•958 
36-4 
35-26 
1-5 
24-9 
6*5 
•913 
47-9 
54-99 
1-2 
37 7 
8-8 
•841 
57-6 
71-24 
1 
49-6 
10-6 
•775 
63*4 
80-03 
0-8 
65 
12-4 
•684 
69-0 
87-22 
•5 
98-7 
149 
•492 
734 
95 02 
•2 
144*2 
175 
•214 
84-9 
99-21 
1 
161'7 
17-8 
•109 
87-4 
99-82 
•05 
170-9 
17-9 
•055 
88*7 
99-97 
•01 
178-2 
18 
•011 
89-8 
99-99 
0 
180 
18 
0 
90 
100 
As we shall see immediately, the above table gives for arcs smaller than 
100° values of times of transit distinctly too high; but for arcs greater 
than 100° the values agree fairly well. This shows that the speed of 
propagation must begin to increase more rapidly with depth than is 
indicated by the assumed formula for v, attaining an almost constant value 
through a large proportion of the inner parts of the earth. 
Let us assume, then, a second case in which the formula 
V 2 = Y 2_^ X 2 
applies from the surface to a depth of only one-tenth of the radius. For 
greater depths the speed is the same throughout. 
After a few trials I chose the expression 
v = 24*4571 -06 -a 2 
as holding true from x = 1 to x — 0‘9. At the surface the speed is 6 kilo- 
metres per second; and at all points in the globe up to value ^ = 0-9 the 
speed is 1223 kilometres per second. 
The ray will be wholly curved when it does not penetrate deeper than 
x — 0*9 ; but when it penetrates deeper than this limit, the middle portion 
of the ray will be straight. 
