174 REPORT OF THE CALIFORNIA EARTHQUAKE COMMISSION. 
disturbance and the free period of the pendulum; the curves are calculated from equa- 
tion (90). Figure 48 shows the phase differences for the same variables calculated 
from equation (88).'_ We notice that for values of « not too large, the magnifying power 
increases with the ratio of the periods to a maximum and then diminishes indefinitely. 
The position of the maximum, found by equating to zero the derivative of (90) with 
respect to P/T,, occurs when 
Ve KT) 
—)|)=1—2 
(7) =e) 2 
a8 Paes 
V1 — (P/T,)* 

and its value is 
W (max) = (93) 
For small values of ¢ the magnifying power varies enormously for different periods, 
becoming very large for periods approaching the free period. Instruments with small 
‘damping emphasize certain periods unduly. As we increase e, W becomes more uniform 
and when e is about 8:1, W varies by less than one-tenth of its value for all periods up 
to the free period, and is very nearly equal to V. This amount of damping would be 
excellent, but it would not make the curves of disturbance and record alike, for altho 
the magnification of the different periods would be practically the same, figure 48 shows 
that the phase differences would not. Nevertheless this offers great advantages; in 
the case of nearly simple harmonic movements, which probably occur not infrequently, 
our record would show the magnifying power without long calculations, whatever be the 
period, up to the free period; and the record would show directly the relative displace- 
ments in different parts of the disturbance, without unduly magnifying certain parts. 
With this value of the damping ratio the proper movements of the pendulum would be 
damped out in one or two vibrations. The longer the proper period of the pendulum, 
the greater the range of periods over which the magnifying power remains nearly con- 
stant. This is the principal advantage of long proper periods when recording harmonic 
disturbances. 
For increasing values of ¢ the position of the maximum moves to the left and becomes 
zero When 1—2(«7',/27)?=0, which corresponds to e= 23:1. For values of e greater 
than this there is no maximum; the magnification is greatest for infinitely small values 
of P/T,, and diminishes for all greater values; when e becomes «0:1; 27/«7', equals 
unity, and the instrument is deadbeat; W is considerably diminisht and varies greatly 
in value. 
The magnifying power for tilts is shown in figure 49; it is equal to the variable part 
of that for displacements multiplied by (n/i)(Q/T,)*. Its value is zero when Q/T, is 
indefinitely small; it increases with this factor and reaches a maximum when 
Q_ 1 
T, 1—2(xTy/2n)* tee 
when its value is 
‘ 7, 
U (max Bi reeed ta), 93a 
3 VQ/Ty 1 sie 
it then diminishes to n/i when Q/T,, is indefinitely large. The position of the maximum 
is at Q/T,,=1 when e=1 (i.e., «<=0); it moves to the right as e increases, reaching in- 
finity when 1 — 2 («T',/27)’=0; ore = 23.1. For greater values of e there is no maximum. 
There is no value of e which produces a fairly even degree of magnification for even a 
? For e = 1:1, the difference of phase is 0.5 for values of P/T7'o less than 1; and is 0 for values of P/ To 
greater than 1, 
