96 REPORT—1858. 
a, b, c, denote the distances between them; let » denote the unknown hori- 
zontal velocity ; and let © denote the unknown angle made by the coseismal 
lines 2 Ax, y By, with the line A B joining the first two stations ; and ¢,, f,, ts 
be the times of the observed shock at A, B, C, respectively. 
Letting fall the perpendiculars p and q, we find, 
+: ba 2 alg aaa 
y= _ 9 sin (B—®) Ape eae 
= na. 
p _esin® 
t.—t, t,—t, 
— 
Equating these two values of v, we find 
c(t,—t,) sin b=a(t,—t,) sin(B—®). 
Expanding, and solving for tan ®, we finally obtain 
a(t,—¢,) sin B bie) 
c(t,—t,)+a(t,—t,) cos B 
Having found ® by means of this equation, we can then determine v from 
either (1) or (2). 
tan o= 
Given the Horizontal Velocity of an Earthquake at any two points, and its 
absolute velocity; to find the position of the focus from which it has 
proceeded, 
m 
Let A and B be the points under consideration, and for simplicity suppose 
them to lie at opposite sides of the unknown focus F, and in the same vertical 
plane passing through F. [These suppositions are only made to simplify the 
figure, but do not in any way diminish the generality of the result. ] 
Let AX be the space moved through on the surface of the ground at A 
in the unit of time, and equal v the horizontal velocity, and let BY be the 
velocity at B and equal v'. Letting fall the perpendiculars AP and BQ; 
PX and QY will denote the spaces described by the earthquake in a radial 
direction (FX or FY); they are therefore equal and each is the absolute 
velocity of the earthquake=V. Hence ' 
cos AXF=”. ° ° ° ° . . . . (1) 
v 
cos BYF=, . of 5 os. Serene ae 
Vv 
Therefore since v, v', V are all known quantities, the angles A X F and 
B Y F are also known, and therefore the lines X F and Y F may be drawn, 
and their intersection F will give the required position of the focus. 
Corol. 1. If the position of the point O, at the surface, from which the 
earthquake appears to radiate, be known; one velocity will 
determine the depth of the focus. 
we 
