1907-8.] Professor C. G. Knott on Seismic Radiations. 219 
2. That the speed of propagation depends only on the distance from 
the centre. 
Let v be the speed at distance r and T the time along any path. Then 
by Hamilton’s general method applied to brachistochronic problems we have 
( 1 ) 
where r, 0 are the polar co-ordinates of a ray passing in any chosen diametral 
plane including the origin of the disturbance. 
Put r = x R, where R ( = 6370 kilometres) is the radius of the globe, and 
T = SR. Equation (1) then becomes 
/STVj-I 
^0T\ 2 
_ 1 
\3r/ 
KrdOj 
v 2 
2 /0S\ 2 cr 
00 / 
(!') 
We get a solution of this by putting 0S /d6 = a, a quantity independent of 
x and 0, and then integrating the equation 
S_ It _ 2 
x V v 2 a 
0S_ 
x m ~ 
between appropriate limits. 
All rays are supposed to begin at the source x = l, 0 = 0. Hence 
R 
= S = a 6 + 
dx 
a 2 
(2) 
an equation which gives the time of transit from the origin to any point x, 0. 
The equation of the path is obtained by equating to an arbitrary 
constant the result of differentiating S with respect to the constant a. 
Hence with the same limits any ray is represented by 
0 = 0 — a 
dx 
l X Jx 2 /v 2 - a 2 
(3) 
each particular ray corresponding to a definite value of the parameter a. 
If \fs is the angle at which the radius meets the ray at any point, we have 
, . 035 
cotan = — - = 
r xdO 
- 1 
G) 
At the surface where x = l, this quantity becomes the tangent of the angle 
of emergence of the ray. Representing the angle of emergence by e, and 
writing U for the value of v at the surface, we get 
tan e 
\/ a 2 U 2 
1 or cos e = aU . 
G') 
When a wave-front impinges on the earth’s surface, the speed of pro- 
