168 REPORT OF THE CALIFORNIA EARTHQUAKE COMMISSION. 
Since this holds for all values of t, we must have 
COs x (75) 
4 nr 

Vé= —<,005(pt —x ne as ny (singe + 5,sin Bp. : -) 
The series converges so rapidly that we may neglect all but the first term; indeed, if we 
attempt to draw the curve represented by the series making the amplitude of the first 
term 25 mm., that of the second term would be a little less than 1 mm. and would have 
a small effect (see figure 45); that of the third term would only be + mm., and its effect 
would hardly be perceptible on this scale. When we consider that the friction is by no 
means constant during a half swing of the pendulum, and that the curve recorded by 
our instrument is by no means an accurately harmonic curve, we feel entirely justified in 
accepting the value of € obtained by neglecting all terms of the series except the first, as 
representing its true value well within the limits of our observations. We then have 


Vé = —<c0s (pt — x) + SEF sin p= B cos (pt — $) (76) 
where 
A : “PP 4 nr eda be ZMbs 16 n*7? 
sae bat sisaas demienes ica Sk op B oats a A sin x + ——— = (77) 
If, however, we wish to take into account the second term of the series in equation 
(75), the second term of equation (76) must be increased by (4n?r/7p’ )(sin 3 pt/27), and 
we observe that tt will have no effect on the maximum amplitude if ¢ is 0, or +60°, or 
+120°, or 180°; that it will increase B by 
Arn?/27 ap? if p= + 30°, or + 150°, or — 90°; 
that it will decrease it by the same amount 
if ¢= —30°, or —150°, or +90° If we 
suppose the period of the disturbance to be 
twice that of the pendulum, n?/p?= 4; and if 
r=0.2 cm., then the change in B may, at 
most, amount to 0.8x4/277, or about 
gis cm.; and if V is 10, the alteration in the 
calculated value of the amplitude of the 
earth’s disturbance may amount to 34> cm., 
or sz mm. As the actual amplitude is apt 
Fie. 45. to be one or more millimeters to produce a 
movement large enough to justify us in 
regarding p’ as a constant and thus make these calculations apply, the effect of the 
second term of the series may be neglected within the limits of errors of observations 
and theory. These data are fair values for the Bosch-Omori seismograph; for other 
instruments they would have to be modified.' 




1 If we wish to avoid all approximations in our solution, we can do so by replacing the two series 
of equation (73) by their values 

RIT ares AR (cospt + 5, 00s 8pt-+- : pee fae i igh Sr iy +} 
2p ™p 2p rp 32 
on integrating we find 
A mn?rt _ n*rt? 
Vé=— = cos t— 
E 2 Cpt — x) $——— op ‘ 
This equals the values given by equation (75) between t=0 and t= P/2; but it does not hold outside 
these values; and the variation from the harmonic form is not so readily seen. 
