180 REPORT OF THE CALIFORNIA EARTHQUAKE COMMISSION. 
coordinates of the CG will be X, Y, Z—1. In the figure, we have omitted the linear dis- 
placements for the sake of clearness, and have represented the CG, as not moved by the 
rotations; this introduces no error as the angular rotations are all given their proper 
values. As in the case of the horizontal pendulum, let us refer the motion of the 
pendulum to three axes fixed in the pendulum and moving with it; and which are 
principal axes of inertia. Axis (3) lies in the line from the point of support to the CG. 
xe 
(1) 

Z, Fia. 565. 
Axes (1) and (2) are the rotated positions of lines at right angles to (8) and passing thru 
the CG, which were, before rotation, parallel to the fixed axes of codrdinates. Axes 
parallel with these and passing through the point of support have primes. 
We assume that there is no rotation around the axis (8). If the pendulum were sup- 
ported at a mathematical point, no such rotation could be set up as the direction of the 
force there passes through the axis; practically the support is a small surface and it 
might be possible for a small moment to exist around axis (3), but it would be so small 
that we may safely neglect it. 
What is actually measured is the displacement of the CG relative to the CG,; that is, 
the angles 6, and 6,; we must therefore form our equations of motion connecting @, and 
6, with the displacements and rotations. Using the same notation as before, except 
that @, and @, are used for the angular displacements of the C@ relative to the CG,, we 
follow the same method to develop the equations of the pendulum. 
The linear accelerations of the CG are given by equations (4), and Euler’s equations 
of angular accelerations around the CG are 
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