188 REPORT OF THE CALIFORNIA EARTHQUAKE COMMISSION. 
Writing z, for the ordinate of the CG,, when at rest and the mass at the end of the rod 
is M, we get from (137) 
6 BS (2,—§— al) = —2 Mgl’ 
The last two equations give by subtraction 
[MJP dz 
ad dl 
(138) 
zis the absolute ordinate of the CG, z, of the CG,; and z,— ¢ is the ordinate of the CG, 
relative to the support. If the support vibrates under the action of earthquake waves, 
€ will vary. In order to express the displacement in terms of the motion of the CG 
relative to the CG, we must move our origin to z,, /, and call the displacement of the CG 
from this point 2’; that is, we substitute z—z,=2', and since z,—¢ is constant when € 
is varying under the vertical movement of the support, d’z,/dt? = d’¢/d?; we get 
es Mi? (5 rt 7B) (139) 

3EJ \d? dé 
Introducing damping and frictional terms, and putting 3 BJ/[M]l’ = (2 7/T,)? =n’, this 
may be written 

eet oe ee 140 
de aa eh aa cdtine a) 
For the equation of the marking point we multiply by m=m,m,- - +; and since ¢, the 
displacement of the marking point equals mz’, we get 
Oe ee ee eee 141 
da nae saa ss, 
which is entirely similar to the equations of the other forms of vertical motion instruments. 
As we have only considered linear displacements, the position of the origin of codrdi- 
nates is unimportant; but if a rotation occurs, it must be considered. A rotation around 
the CG, as origin would evidently have no effect, if we neglect the moment of inertia 
of the mass about its CG, as we have done. But an angular acceleration d*w,/di? around 
an axis through O at right angles to the paper would make z, — §— @,l instead of z,—¢ 
constant during the motion and would therefore add a term ld?w,/di? to (140) and 
mldw,/dé to (141). These terms in general would probably be unimportant. 
We see thus that the approximate equations of all forms of seismographs referred to 
the CG, are of the same general form; except that no rotation is present in the equations 
of the vertical motion instruments. The formulas (79a) and (81) are applicable to them 
all to determine their magnifying powers. 
SEPARATION OF LINEAR DISPLACEMENTS AND TILTS. 
A rigid body can be moved from one position in a plane to any other by means of a 
linear displacement and a rotation; altho the direction of the axis of the rotation and the 
amount of the rotation are determined by the two positions of the body, the distance of 
the axis is not; we can choose this distance arbitrarily and then determine the linear dis- 
placement to correspond; and the total displacement of a point of the body will be the 
displacement due to the rotation around the axis plus the linear displacement of the axis; 
as the rotation is independent of the distance of the axis, the nearer the latter is to the 
body, the greater will be the displacement due to the displacement of the axis and the 
less will be that due to rotation; and there is one distance of the axis for which all the 
displacement may be expresst as a rotation. We see therefore the origin of the difficulty 
