INTERPRETATION OF SEISMOGRAMS 57 
determine the corresponding angle e. For the longitudinal 
effect P we have 
cos ¢= V, ae 
and we also have 
V, (i -sin é\! 
cos @é= V. ( 5 ) 
where é is the apparent angle of emergence. 
Now if the rays travel in a straight line from E to A the 
angle of emergence e would be simply 4/2R = 0/2. 
The table, page 54, shows at once that as we proceed to in- 
creasing distances the value of e obtained from the time curve 
is much greater than the corresponding value of 0/2. Thus 
the rays dip more deeply into the earth than does the straight 
line from focus to station. The rays must on the whole be 
concave towards the surface, and we have now to abandon the 
hypothesis that the earth is uniform, and instead to assume 
that the velocity of propagation depends on the depth. Ac- 
cordingly the next step is to suppose that the earth is made 
up of concentric uniform spherical shells, but that the velocity 
v varies as a function of 7 the radius of the shell. On this 
hypothesis the brachistochronic paths are still plane curves in 
planes containing the focus, Earth’s centre, and the station, but 
are now curved, each curve being characterized by the well- 
known equation f/v = c(a constant) where / is the perpendicular 
from the centre of the Earth on the tangent to the curve at any 
point. From the values at the surface we get 
COSA an ale 
Tn an Amn aon 
Now the path is symmetrical, so that if the greatest depth for 
the ray is %,,, the velocity at that depth is given by (R — 4,,)/c. 
If we put 7/v=7 we find that 4 and T are expressed as in- 
tegrals, viz. :— 
b Hil 
A et — 1 c| (GEE) Fi log r dan 
b 2 EN 
2 | @-2) Des ee 
where 6=R/v,. 
Dig—c— 
