152 REPORT OF THE CALIFORNIA EARTHQUAKE COMMISSION. 
The reactions of the support have been replaced by a single force F applied at 0’ 
and a couple. The forces of the couple both pass thru the axis of rotation and there- 
fore can not have a component around it, or around a parallel axis thru the CG. Of the 
components of F, F, passes through the CG; F, is parallel with (3), and therefore F, 
alone is capable of exerting a moment around (3). Similarly only the /, component of 
f can exert a moment around (3). If the latter force is exerted at a point distant J, 
from 0’, we find 
3=FI—fi(,—l) (8) 
Let us replace d*u by d’?(@+.,), which expresses the angular acceleration in terms of 
the acceleration of the pendulum relative to the support, and the acceleration of the sup- 
port; with these substitutions the equation of angular acceleration (5) becomes 
Pt) = FIA ©) 
We must now replace F, by its value in terms of the resolved parts of F,, F,, F, in 
the direction of (1), and then the values of these latter quantities must be obtained from 
equation (4). 
We have 
F, = F, cos (x, 1) + F, cos (y, 1) + F, cos (2, 1) (10) 
Since the rotations are the same for all points, we can determine the cosines of the 
angles in the above equation, by assuming a sphere of unit radius 
with center at 0’, and determining the displacements of the axes 
#“, on its surface as a result of the rotations (w’s) and the relative 
. angular displacement (@). These values follow directly from figure 
Faq. 87, 37, where the points represent the intersections of the axes with 
' the surface of the sphere and the lines represent the displace- 
ments of these points. 
()) 
cos (a, 1) = cos(w, + 6) =1 -§ 
cos (y, 1) =sin fo, +6(1—5)}=0, +0 (11) 
cos (z, 1) = —sin(w, +76) = — (w, + 76) 
All the angles are small, and 2 and @ are considerably larger than the w’s; we have there- 
fore neglected squares of the o’s, products of w’s and @, and 770; but 7@ is an important 
term in our equation; and since 6 is of the same order, these terms must be retained. 
Substituting the values of these cosines and the values of F,, F',, F., from equations 
(4), in equation (10) we get 
da 02 dy § dz " 
r= (a4 le i.) (1 = 4 + (4 a =f) (w, + 6) — (41 tat Mg) (w, + 18) (12) 
‘The codrdinates of CG, are X, Y +1, Z—il; the codrdinates of C@,, during the dis- 
turbance, are 
€é+X+ (Z— il) wy, — (Y+ l) w, 
n+Y¥+1+ Xo,—(Z— il), (13) 
€+Z-—i+(¥+4+/)o, — Xo, 
and the coordinates of CG during the disturbance are 
e=f+ X+(Z—il)w,—(Y+)o,—10 
y=n+(¥+) + Xo, —(Z— io, (14) 
2={+(Z—il) +(Y+Do,—Xe, 
