229 
1907-8.] Professor C. G. Knott on Seismic Radiations. 
respectively by the arcs into which Oa divides the semicircle. The semi- 
vertical angle of this cone is equal to 90° — e, where e is the corresponding 
angle of emergence belonging to the ray OA ; and the area on the spherical 
surface defined by the arc Oa is proportional to 
1 + cos (180° - 2e) = 1 - cos 2e. 
This number, divided by 2, the value when e = 90°, represents the fraction of 
the energy which is finally distributed over the surface defined by the arc 
OA. Thus in the particular case which has been the subject of discussion, 
80T per cent, of the whole energy is found distributed over the compara- 
tively small fraction of the surface whose boundary lies 21°6 from the 
epicentre. In spherical distribution only 3 ’5 per cent, of the whole energy 
would have appeared over this surface. Glancing back to the table for 
Case II., we see that 50 per cent, of the whole energy is distributed over the 
small surface whose boundary lies about 7° from the epicentre, and that 
75 per cent, is distributed over the surface which does not extend to 18° 
from the epicentre. In spherical distribution these surfaces would have 
received respectively only J and 2J per cent, of the whole energy. 
It is interesting to compare the two cases figured side by side on the 
diagram, and to notice how much more concentrated the energy is in the 
neighbourhood of the epicentre in Case II., which is characterised by a 
rapid variation of speed of propagation within the upper layers of the 
earth. 
In these calculations I have, for simplicity, assumed the origin to be at 
the surface. This is never quite the case in large, world-shaking earth- 
quakes. These originate at depths which may vary from 10 to 50 miles. 
Nevertheless, because of the curving of the seismic rays the energy will be 
distributed unequably in a manner similar to what is here indicated. The 
deeper the origin the less unequable will the final distribution be ; but so 
long as the origin lies within the layer of changing velocity, there must be 
the curving round of the seismic rays, carrying their energy with them. 
Let there be two earthquakes of equal intensity but with their origins 
at different depths. The one with the shallow focus will have its energy 
strongly concentrated towards the surface regions immediately in the 
vicinity of the epicentre ; while the energy associated with the deeper focus 
will be less unequably distributed over the whole surface. The latter will 
be registered all the world over as a world-shaking earthquake, while the 
former may appear much more limited in its sphere of action, simply 
because of the small intensity of the tremors which pass to distant 
regions. 
