GENERAL DYNAMICAL THEORY OF SEISMOGRAPAS 5 
L 9 9 I\9 949. —i 
7 PNG PY + 4ep 
represents the “magnification” of the amplitude of the earth 
movement. 
Terms representing the free motion will also appear on the 
record, and these have a diminishing amplitude. Now the 
practical problem is to determine the earth movement + 
from the recorded movement so that even this simple case 
shows us how important it is that the “free” terms should be 
made to subside with rapidity. A fortéord it is evident that if 
« is undergoing complicated changes, it is difficult to form any 
true conception of the earth movement from the seismogram 
unless the “ free” terms are made to subside quickly. Thus 
the necessity for a large value of ¢, that is very great damping, 
becomes apparent. 
The expression for the magnification may be written 
EN) 
where U ={(u? — 1)? + 4u%e?/n?\4 and u=2/f. 
Thus U is unity when p= co that is for infinitely rapid vibra- 
tions, while U is infinite when /=0. Thus the magnification 
is mz¢ for infinitely slow vibrations. 
U clearly becomes a minimum for different values of z 
when 
w=1- 26/2 
and we may choose e/z so as to get the minimum for any 
prescribed values of z. 
If e/z=1/2* we get #=0 as the minimum, and this value 
has the advantage of making the magnification for rapid waves 
more nearly constant for different periods than would other- 
wise be the case. 
If the instrument is aperiodic e/z=1 and there is then a 
minimum at «=o for U, which now takes the form (w? + 1). 
In certain theoretical investigations it is convenient to use 
quantities related to ” and e as follows :— 
h=eln, p*=1-h* and y=(n? - €)* = un 
so that 4=1 or uw? =0 expresses the condition for aperiodicity. 
