THEORY OF THE SEISMOGRAPH. 147 
propagation is cos «.dy/dz, as will readily appear from figure 34. A line making a vertical 
angle « with the horizontal direction of propagation would be turned thru an angle 
cos? «.dy/dz, but this line does not interest us, nor does the corresponding line in the 
case of horizontal displacements, which would have a rotation of cos «.dy/dz. 
These conclusions depend on the assumption that the support of the seismographs 
has exactly the same motion as the underlying rock, or that the column supporting the 
pendulum is fastened rigidly to the rock; if, however, the seismograph rests on a pier, 
even tho it be connected rigidly with the solid rock, the case is different. The movement 
is communicated to the base of the pier, and as its sides are subjected to no constraining 
forces, the top of the pier, in the case of horizontal displacements, would probably rotate 
around a vertical axis nearly like a rigid body, thru an angle equal to the average rota- 
tion of all lines in its base; that is, thru an angle », such that 
1 (270A 120A 
p= — —— cos’ a+ da = 
3 
23 r on EX (8) 
or half the maximum rotation in the solid rock. We assume that the natural period of 
the pier for rotational vibrations is so much shorter than the period of the earthquake 
wave that it does not exert an appreciable influence on the amount of the rotation; this 
assumption seems entirely justified. 
In the case of vertical displacements the pier would be tilted thru an angle equal 
to the tilt of the rock, but its top would also have quite a large linear displacement. If, 
however, the pier were very long, its period might be comparable to that of the shorter 
earthquake waves, and the instrument would record movements which would be a com- 
bination of the movements of the ground with the proper movements of the pier. It is 
probable that the movements of high chimneys and tall buildings would be materially 
affected by their natural periods of vibration. 
Let us now consider waves of condensation, like sound waves, where the direction of 
the displacement, £, is the same as that of propagation, x; the equation of the wave will still 
have the same form as heretofore. A line in the direction of propaga- 
tion or at right angles to it, horizontal or vertical, will have no 
rotation; a horizontal line making an angle « with z will suffer a 
difference of displacement of its two ends equal to dé, or dé sin « at 
right angles to its length; its length is dz/cos«; therefore its rotation 
is sinecosa-d&/dx; the maximum value of this is 7A/\, when 
«=45° and when dé/dz is a maximum. This is a rotation around 
the vertical axis. A line making an angle a on the opposite side of 
the line of propagation is rotated in the opposite direction; it is 
probable that the top of the pier would not rotate at all about 
the vertical, when the base is subjected to this kind of motion. 
Observers have not so far succeeded in directly measuring rota- 
tions; and as we should expect them to be extremely small, 
we shall so consider them until further evidence shows them to be larger. 

ibyey chr 
FORMS OF SEISMOGRAPHS. 
The forms of instruments which have proved practical for recording very small dis- 
turbances are: the ordinary pendulum, the horizontal pendulum, and the inverted pen- 
dulum. The first form is too familiar to need any explanation; the second is a frame or 
a bar carrying a heavy mass, supported at two points nearly in a vertical line, as a door 
is supported by its hinges; so that its position is affected by a small displacement of 
the support at right angles to the direction toward which it points; the inverted pendu- 
