THEORY OF THE SEISMOGRAPH, 179 
The two points of support and the center of gravity lie in a vertical plane when the 
instrument is in equilibrium. The direction of the forces F and P ean be somewhat 
controlled by the direction of the points and of the cups, but friction will alter the 
direction of the forces to some extent. Usually a plane surface is used instead of one 
of the cups, which renders it unnecessary that the distance between the points should 
be exactly the same as the distance between the ceriters of the cups. Taking moments 
about the points of support, we find 
F= a as jose La eel ate (104) 
where the meanings of the letters are shown in the figure. These forces become equal 
when 1, =1'/2, and they make equal angles with the vertical; they then pass thru the CG; 
they become smaller as /’ becomes larger in comparison with 1. When the lower point 
presses against a vertical plane agate surface, the direction of P is horizontal, 1,=/, 
and 
) 12 2 
P= wyrrt aul Pa wi (105) 
Te desl ei = 1AlP. 
(4) The Zéllner suspension (figure 54); the beam is supported by two wires m, and m, 
fastened to the support, one above and one below the beam. The direction of the forces 
must pass thru the vertical thru the CG of the beam; and therefore the angle «, must be 
greater than the angle «,; but the values of these angles can only be found thru an equation 
of the fourth degree, and can only be expresst by a very complicated expression. The 
Zollner suspension has the great advantage of not having any pivots, and therefore, if 
an optical method of registration is used, there is no solid friction. For very slow move- 
ments it would answer very well, but for more rapid movements its motion is too compli- 
cated. It can have linear displacements parallel with and at right angles to the beam, 
as well as a rotation around a nearly vertical axis at right angles to the beam. The 
linear movement parallel with the beam also caused a vertical movement of the mass. 
These various movements, themselves the effects either of linear displacements or tilts 
of the support, could not be separated from each other by a single registration; and it 
would be impossible to interpret the record. ‘To avoid these complications Prince Galit- 
zin has proposed to have the beam press by a point against an agate plate placed close 
to the axis of rotation, and he has shown that even when the pressure is very light, the 
device will prevent the first two movements. Another way would be to fasten the point 
of the beam where it crosses the axis of rotation by guy-wires. They would prevent it 
from moving out of this position, but would not interfere with small rotations. Prince 
Galitzin has suggested this method for other instruments. Either of these devices pre- 
vents all relative motion except a simple rotation, without introducing friction, and the 
theory of the instrument then becomes the same as that already given for the Gray or von 
Rebeur-Paschwitz forms. All instruments of the Zollner type in use up to the present 
time have no device to prevent the complicated motions, and in attempting to interpret 
the records of the California earthquake as given by instruments of this type, we must 
assume that only rotations take place. 
THE VERTICAL PENDULUM, 
Let us now consider the movements of an ordinary vertical pendulum whose support 
is subjected to an earthquake disturbance producing the three displacements, &, 7, §, 
and the three rotations, ,, o,, o,. 
Let O, figure 55, be the origin of codrdinates and let X, Y, and Z be the coérdinates 
of the point of support; then if J is the distance from the point of support to the CG, the 
