M14 REPORTS ON THE STATE OF SCIENCE, ETC. 
globe with O and D as centres, intersecting in two points one of which is E. 
But drawing these circles on a globe is not a very accurate operation, and it is 
convenient to calculate, if it can be done quickly, in what direction outwards 
from O or D to look for the epicentre. 
Cc 
M 
© 
Let EO—ED=2z2: and put EO+ED=2A 
Then 2a must be less than the distance OD (say 2d) between the stations; and 
approximately OM=2z, OD=2d, so that cos EKOD=2/d, which gives 
approximately the angle made by EO with OD. 
The point to which attention is here called is that this simple equation 
(which is, moreover, independent of A and thus readily tabulated) gives with 
considerable precision the angle ECD, where C is the midpoint of OD. This is 
true either for spherical or plane geometry. Consider first the latter. We have 
2 EC. d. cos ECD=EC?+d?— ED*?= KO?— EC?—@? 
.. 4 EC. d cos ECD=EO?—ED?=4 Az 
ns 2 (EC?+d?)=E0*+ ED? =2( A?+2%) 
ECD=". = 
cos EC d° (42a! 
Since the equation is only likely to be required when z and d are small compared 
with A, the approximation is close. 
For spherical geometry the equations are 
sind. sin EC cos ECD=cos ED—cos EC . cos d=cos EC . cos d—cos EO 
2 sind. sin EC . cos ECD=cos ED—cos EO=2 sin A. sin 2: 
2 cos d cos EC=cos ED+cos EO=2 cos A . cosa 
Thus cos d sin KC=(cos? d—cos? A cos? x)3 
tan x sin A 
and cos ECD= : 
tand (cos? d—cos? A cos? x)! 
_ tan x sin A 
tan d (sin? 4 cos? a+sir2 x—sin®? d3 
where the approximation is clearly of the same order as before. 
It may be convenient to use a flat projection of the sphere. Thus we may 
take O as the centre of a gnomonic projection, so that a circle of radius 7 
round O is projected into a circle of radius tan 7 on the flat. The angles round 
C will then be projected angles, but if C is not too far away from O, the error 
made in setting them off uniformly round C will not be large. 
There is another method of finding the points of intersection E of the two 
circles, which may be useful as a check on the former. Let the constants for 
O and D (as given on p. iii of the ‘Large Earthquakes for 1916’) be (a, b, ¢,) 
and (a, 6, c,) : and let E be denoted by (A, B, C). Then 
cos LO=a,A+b,B+e,C 
cos ED=a,A+6,B+¢c,0 
a 
