FINITE FOCAL DEPTH REVEALED BY SEISMOMETERS. 
47 
The quantity e is called the apparent angle of emergence at the surface, and is defined 
by 
tan e = Z/H 
where 
Z is the observed vertical component of displacement 
and 
H is the observed horizontal component of displacement. 
Thus e is the direct subject of measurement by vertical and horizontal seismometers. 
The angle e is called the true angle of emergence of the brachistochronic ray. It cannot 
be directly measured, but may be calculated from the time curve for P by the formula 
cos « = V, Ta , 
where Vi is the speed for longitudinal waves at the surface, and T the time is supposed 
expressed in terms of the epicentral distance A. 
When the conditions of reflexion are examined it can be shown that for a longitudinal 
ray incident 
cos e — 
Vi 
V 2 
{* 
(1 — sin e)}\ 
where V 2 is the surface speed of transversal waves. 
It is seen from the table that the values of e calculated from Zoppritz’s curve do not 
agree with the values of e directly measured at Pulkovo. The discrepancy is so marked 
that we may set aside the supposition that the Pulkovo values are merely instrumental 
errors. In a matter so important Galitzin would hardly have published them if he had 
not felt assured that they were substantially correct. There remain two alternatives : 
(1) that the ratio Vj/Vg for Pulkovo depends on the angle of impingence in such a way as 
to exactly annul the discrepancy. The probability of such compensation of actual facts 
to explain a theoretical formula must be regarded as small, and so we are left with 
alternative (2) that within the limits of possible error in the time curve we can modify 
it so as to agree with the direct measures of e. We shall show that this alternative is 
quite possible within quite a large range of A. But we must at once point out the 
somewhat startling consequence of accepting the Pulkovo numbers as correct. 
It has been shown that a ray which emerges with a minimum angle must have set out 
from the focus in a direction at right angles to the radius vector from the earth's centre 
to the focus. Thus for a minimum angle at A = 4000 km. we find that even for a 
uniform earth the depth of focus required is about 0-2 of the earth’s radius, or about 
1250 km. The actual value may be a little less or a little more, according to the way 
in which speed varies with the depth. Anyhow, this is a much larger estimate of depth 
than has formerly been suggested, viz., of order less than 100 km. 
h 2 
