CALCULATION OF MOTION OF GROUND 93 
In the equation —x=y+2kai+ paz since p? =42?/T? where T is the natural 
period of the seismograph, when undamped, we see that the correction pa, 
which must be added to y to get the true motion of the ground is inversely as 
T? for all values of the time ¢. If T is very large this correction may be negligi- 
ble. In the same way the damping correction 2a, is proportional to the damp- 
ing coefficient. Thus for an undamped seismograph of very long period —x =y 
so that the seismograph gives the ground motion exactly. Any increase of the 
damping and any reduction of the period T increases the corrections and so 
increases the difference between the seismograph deflections and the ground 
motion. If k= the seismograph is critically damped. 
To find a; it is convenient to make a tracing of the seismogram on graph 
paper and draw a curve for the area by counting the squares between the y 
curve and the zero line y=0. The zero line is usually not marked on the seis- 
mograms so that it is necessary to draw it. A small error in the position of the 
zero line makes a large error in a. Thus if it is drawn at y = —3 instead of at 
y =0 we get a; too large by 3¢. Such errors usually occur but are easily allowed 
for when it is remembered that a:=0 for large values of ¢. If a; does not fi- 
nally approach zero as ¢ increases then by slightly shifting the zero line it is 
made to do so. Very often it is found that the area curve fluctuates about a 
stiaight line through the origin finally approaching this line. In such cases it 
is clear that the zero line is too high or too Jow. The line through the origin 
may then be taken to be the zero line for the area. The integral a2 is then ob- 
tained from the curve giving the area in the same way. If it does not finally 
approach zero it must be made to do so by a suitable shift of its zero line. If 
an error Aa is made in the area a; at time f, then this produces an error 
Aa(t—t,) in the value of a2 at a time ¢ later than #,. Such errors are usually 
appreciable and cause the curve obtained for this integral to drift up and 
down. It is therefore usually necessary to draw a new zero line for this curve. 
This zero line must be a smooth curve such that the value of the integral 
oscillates more or less equally above and below it. 
Instead of endeavouring to correct such errors by shifting the zero lines, 
a process which depends a good deal on arbitrary judgment, one may carry 
out the calculation of x using the uncorrected curves for the area and the 
integral. When this is done the curve obtained for the motion of the ground 
usually appears to oscillate about a smooth curve which may drift very far 
from the zero line. This smooth curve may be drawn and taken to be the 
true zero line for the ground motion. These methods of correcting the calcu- 
lated ground motion are necessary unless the areas are very accurately de- 
termined. They obviously result in the removal from the ground motion of 
any long period oscillations having periods long compared with the period of 
the seismograph. The constant is nearly equal to 27/T unless the damping 
is large, T being the periodic time of the seismograph. In getting the areas of 
the curves it is convenient to use a time scale such that p=1. 
Before considering results obtained with seismograms some simple the- 
oretical cases will be considered. Suppose y=0 when ¢<0 and y=a sin pt 
when ¢>0 and that k=0. In this case 
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