154 REPORT OF THE CALIFORNIA EARTHQUAKE COMMISSION. 
with P = 20 secs. and A = 5 mm., this has a value of about 0.5 mm. per sec. per sec. This 
is the order of the terms d?&/dé, d?n/di?, d?¢/di?; the last is very small in comparison with 
g, Which is nearly 10,000 mm. per sec. per sec., and may therefore be neglected. 6°/2 
is small in comparison with unity, but it is of the same order as 10; nevertheless d?&/di* 
is so small in comparison with g, that we may neglect (67/2) (dE/dt*) also; d?n/dé* is of 
the same order as d°&/dé, but it is multiplied by (#,+ @), which with some instruments 
may amount to 74; if the accuracy of our measures is not greater than this, we may omit 
this term in comparison with d*&/di?. The product f,(1—67/2)=f,. The left-hand 
member of the equation may be written Ml'd?0/di? + 1,(d?0/di? + d’w,/dt*) ; the omission of 
d’w,/dt” is equivalent to substituting, in the second term, the angular acceleration rela- 
tive to the support for the absolute angular acceleration. Since the maximum value of 
Os is of the order of 5 x 10-7 and the maximum value of @ is of the order of 5 x 107%, and 
since they would have the same period, we find that d?6/di? would be about 1,000 times 
as large as d*w,/di?; and since in general I, is much smaller than M?, it is clear that 
we make no material mistake in omitting Po,/ di?. Our equation then takes the form 
19 ap =m M —9(4 +0) ce 
In the undisturbed condition the CG lies in the vertical plane containing the axis of 
rotation, this axis making a small angle 7, with the vertical. When the instrument is 
disturbed the position of equilibrium is in the vertical plane containing the axis of rota- 
tion in its disturbed position. Using the same device as on page 152, we see by figure 38 
that the angle thru which the plane of equilibrium is turned 
Se : z is — (w,—1,)/i; but since the support itself is turned about 
ate the vertical thru an angle ,, the angular displacement of 
Fie. 38. the plane of equilibrium relative to the support is 
— (w, — 1@,)/1— @,, which reduces to —@,/1. The last term in equation (20) is there- 
fore the moment due to gravity tending to bring the pendulum back to its position 
of equilibrium, and it is proportional to the angular displacement from the position of 
equilibrium. 
The value of the new angle 7, between the vertical and the axis of rotation, reduces 
practically to 7,—@,, on account of the small angle thru which the plane of equilibrium 
has been rotated (see figure 38). Since , is of the order 5x 1077 and 7, for the von 
Rebeur pendulum, where it has a smaller value than for any other instrument, is about 
1:700, we see that its value is about 2: 3000; for other instruments it is still smaller; 
we may omit », and consider that the inclination of the axis of rotation to the vertical 
has not been changed by the disturbance. 
The equation contains f,l,, the moment due to the reaction between the pendulum 
and the indicator. Its value can be determined from the equation of the indicator and 
then substituted in equation (20). The indicator is itself a small, horizontal pendulum 
and is affected by the disturbance; its general equation will be of the form of equation (16). 
Let us assume that the axis of rotation of the indicator and its cg, lie in the axis of y 
thru the CG, of the pendulum; the codrdinates of the cg, then become 0, 1,, 0 (see fig. 36) ; 
putting these values for X, Y +1, Z—v7, in equation (16) and writing primes to mark 
the quantities referring to the indicator, its equation becomes 
ee ei [eee Jie | Sa7, i he Gar at?) ] os 



® dt lt? dt? de ° MU" dt * at 
f,| has a positive sign because the force is applied so that a positive force causes a 
positive angular acceleration; we have assumed 71= 0, and that the reaction between the 
