THEORY OF THE SEISMOGRAPH. 151 
1, the inclination during the disturbance ; 
6, the angular displacement of the C@ relative to the support, the positive 
direction being the same as that of o,; 
l, the perpendicular distance from the CG to the axis of rotation at 0’; 
X,Y,Z, the absolute codrdinates of 0’ before the disturbance; 
x,y,z, the absolute codrdinates of the CG at any time; 
F, the force applied at 0’; 
ei aa its components parallel to the fixed axes and to the moving axes, respec- 
heey ee ep tively. 
the components of the force exerted on the pendulum by the indicator; 
the mass of the pendulum; 
I,, I,, Iz, the moments of inertia of the pendulum about the principal axes of inertia 
through the CG; 
—,7, 6, the linear displacements of the support due to the disturbance; 
@y) Oy, 3 the angular displacements around the axes, due to the disturbance, the posi- 
W135 @y) Ws, tive directions being indicated in the figure. 
For the sake of clearness the displacements of the support are not shown in the figure, 
but they can easily be imagined. 
The linear accelerations of the CG are given by the equations 
ax dy 
M —= Hh a M —~ = Ff M 
ag tats ap uth 

2 = F,— Mg (4) 

In order to see exactly what approximations we make, we must use Euler’s equations 
for moving axes to determine the angular accelerations; the motion is referred to the 
instantaneous position of the 3 principal axes of inertia thru the CG, which we have called 
(1) (2) and (3) respectively; as we only observe the rotation around (3) we may neglect 
the equations referring to the other axes; the equation is 
Ue cp 7 or oe _ 
Js dt? (h— 2) dt dt Cs ) 
where p» is the absolute angular acceleration around the instantaneous position of the 
axis (3), and C, is the moment of all forces around this axis. As the pendulum has no 
relative motion around the axes (1) and (2), its angular velocities around their instan- 
taneous positions are the same as those of the support. Since the support is supposed 
to move with the underlying rock, its angular displacement will be the same as that of 
the rock, and will be given by equation (2). Its angular velocity will be obtained by 
differentiating this equation with respect to the time, we thus find: 
doy » _(27)2A . 
fae) tse AP (6) 
and with the values there used: A=5 mm., P= 20 secs. ; \ = 66 km., this becomes about 
3x 1077; and dw:/dt has a value of the same order. We may write (as we shall see 
further on) 

2 2 \2 
p= COs (3) “ia (maz) = (=) ® (7) 
making P = 20 secs. and ® = 0.005, which is probably a smaller value than it would have 
under the assumed disturbance, we find the maximum value of the relative angular 
acceleration of the pendulum to be about 5 x 10~*, a quantity far larger than the product 
of the two angular velocities given above. We may, therefore, without appreciable 
error, neglect the second term of equation (5), 
