THEORY OF THE SEISMOGRAPH. 157 
In order to determine the value of linear displacements, we must either neglect the 
rotation as small, or determine its value as a function of the time from some other instru- 
ment; and then either integrate the equation, which can be done if it is found by the 
record to be a simple form, say a simple harmonic curve; or we must laboriously meas- 
ure from the record the successive values of d?0/dt?, d@/dt and 6, which when introduced 
into the equation will give us the successive values of d?&/dt?._ A double summation of 
these values will then give us the successive values of the displacement £ So far as I 
know this process has only been carried out once and then without taking into considera- 
tion the constant p’.1| The process is very laborious and emphasizes the advantage 
which some other form of instrument would have, in which the relation between its dis- 
placement and that of the earth would be more direct and simple. 
DETERMINATION OF THE CONSTANTS. 
But with the instruments we now have, it is important to determine the values of 
these constants, which can be done as follows. If the support were subjected to a very 
rapid but small movement, the second derivatives would be so much larger than the other 
terms in the equations (25) and (26), that the latter could be neglected and we should 
have 
peo Pa nl @é 
=e ig o7 
ar de df) iL dé a9, 

Integrating and neglecting the velocity multiplied by the time of the movement, as 
the latter is supposed extremely short, we get 
L (6-6) =£—-& a— w=" (—&) (28) 
This shows that for a movement of this kind a point on the pendulum distant L from the 
axis of rotation will have a relative displacement equal, but in the opposite direction, to 
that of the support; that is, that it will actually not be displaced by the movement. 
This point is the center of oscillation. It is also the point at which the whole mass of 
the pendulum might be concentrated without affecting its motions; L is therefore called 
the length of the mathematical pendulum of the same type; such a mathematical pendu- 
lum would have the same period as the actual pendulum (as we shall see later), but 
we must remember that ZL, as defined here, is not the length of a simple pendulum 
having the same period as the horizontal pendulum. 
We also see from the second equation that the actual movement of the marking point 
will be ni/L times as great as that of the support; this then will represent the magni- 
fying power for small rapid linear displacements, and we may represent it by V. Its value 
is evidently 
1a ale ly L_nl ml, 
Fae pa 2a A ee 29 
yeep Oo Ti es a) 
if we write m=m,m,m,- - - where m,=I,'/l, - - - ete., i.e., m, equals the ratio of the 
long to the short arm of the first lever, etc. If on the other hand there is no linear dis- 
i Ze A tind. ot Ears termern ye 
oe ie ok ga dc be hates en ea ee eee ees RS 
2 <—-—-— £ —-—-- —- > 
Fig. 39. 
placement, but a small rapid angular acceleration around the axis of y, the pendulum 
is not affected at all; for the equation does not contain the angular acceleration. This 
1 By General H. Pomerantzeff, Recherches concernant le sismogramme tracé a Strassbourg le 
24 Juin, 1901. Acad. Imp. Sci. St. Petersburg; C. R. Com. Sism. Perm., 1902, Liv. 1, pp. 185-208. 
