86 REPORT—1858. 
their local emplacement, it remains to describe the instruments that appear 
to me the best calculated for the attainment of the objects we can at present 
propose to ourselves in seismometry, and to point out how such may best be 
applied ; as also some indirect methods of arriving at the most important and 
interesting primary result, that we are entitled to expect in the first instance 
from such researches, namely, an approximation to the actual depth of focus 
within the earth, from which earthquake-impulses are propagated to the 
surface, 
Were it possible to construct a perfect seismometer, it should record 
simultaneously, Ist, the movements, both horizontal and vertical, of the 
elastic wave itself, viz., the excursion or amplitude, the altitude, and the 
maximum velocity in the coordinates a, y, and z,—z being vertical ; 2nd, the 
movements of translation of the “advancing form” or wave itself at its 
emergence upon the earth’s surface, with the velocities in the correspond- 
ing coordinates 2%, Y2, and z,. 
These involve alone twelve equations of condition ; and we assume that 
the elastic medium (the earth) through which the waye is transmitted, is 
homogeneous, in density and elastic modulus; and that the final wave- 
movements, of the free outlying stratum at the surface, obey the same laws 
as do those of the successive “ couches ” beneath, 
Generally, we must assume the elasticity perfect, and that the vis viva of 
any particle in motion, Am, is determinable from its velocity at its position of 
equilibrium. From the general equation of wave-motion 
v= 4 COS (= (w—at)), 
we have the velocity at any point where a? is the intensity, \ the amplitude, 
a the transit-rate or velocity of propagation, x the abscissa, and ¢ the time. 
At the position of equilibrium v=a, and the vis viva of the particle Am 
during the whole undulation is Ama’, and proportionate to a. The wave 
we must suppose emanating from a central point, and propagated outwards 
in all directions alike, in imaginary, concentric spherical “couches.” The 
vis viva must remain constant during the whole propagation. The velo- 
city of propagation @ is also constant ; and the mass of the medium in wave- 
motion at any moment of the translation is the same; so that, if r=the 
radius of any such spherical ‘ couche,” the work done in it by the wave 
is proportionate to 7?xa*, and constant for the whole transit, a? being 
oc ye As, therefore, the mass in simultaneous undulation is constant, the 
V4 
thickness of each imaginary successive “couche ” must decrease as 7? ; and 
so the displacing power of the wave diminishes also as r*, and the work done 
by the wave within any such “couche” of determinate thickness=25Ama?, 
—or M, being the mass in simultaneous undulation, = 3Ma?. 
The wave at its origination, starts in any radius, with one normal and two 
transversal vibrations, the separate determination of which wouid require a 
corresponding increase in the number of equations for w, y, and z; and in the 
recorded facts by the instrument. It is obvious, then, even with the utmost 
simplifications we can assume as to the molecular condition of the medium 
(the earth), that practically we must be content with a seismometer that shall 
record only some of the more important conditions of the earth-wave, and in 
such a manner as shall enable us, indirectly, to arrive at others. And in 
considering the relative importance of the several elements, the maximum 
velocity of the wave at its point of emergence upon the surface, with the 
