90 H. A. WILSON 
because of the small value of the ratio of the maximum value of y to that of 
x. A critically damped seismograph with a period very small compared with 
that of the waves in the ground gives a deflection very nearly proportional 
to the acceleration of the ground. The equation 
1 t 
y= — fo etevsin ult — Fan 
XL 0 
when »=0 for critical damping becomes 
t 
y= fo rend — F(o)dr. 
0 ° d 
If the period of the seismograph is very small so that & is large then the fac- 
tor e~*“- is negligible except when 7 is nearly equal to ¢. If we suppose that 
F(n) varies slowly and so can be regarded as constant during the short inter- 
val when 7 is nearly equal to ¢ then 
t 
y = FY il een — n)dn 
0 
or 
F(t) 
yee — « *(1 + #)) 
or when ¢ is not very small y = F(¢)/k?. Thus in this case since F(#) = —Z we 
have y= —#/k?. 
If a very short period critically damped seismograph can be made suffi- 
ciently sensitive, it will indicate the acceleration of the ground with consi- 
derable accuracy. Whenever a sudden change in the acceleration of the ground 
occurs, the change in the seismograph deflection will be proportional-to the 
change in the acceleration multiplied by the factor 1—e~*‘(1+&#) where # is 
the time since the sudden change took place, which very soon becomes equal 
to unity when £ is large. 
Now consider the deflection of the seismograph due to a single wave. Let 
x=A(1—cos wt) from t=0 to = 27/w and x =0 when 1<0Qand when t>2z/w. 
In this case y is given by 
Aw? t 
y= —-— e—*(t—®) sin w(t — ) cos wydn 
B 0 
from t=0 to t=27/w and by 
Ao? 2r/w 
y= - e—*(') sin u(t — ) cos wndn 
td 0 
when ¢ is greater than 27/w because the acceleration is zero when t> 27/w. 
First suppose that the seismograph is undamped so that k=0 and then 
when t>271/w we have 
Aw? 2r/w : 
y= —— sin u(t — 7) cos wndn 
ld 0 
