156 REPORT OF THE CALIFORNIA EARTHQUAKE COMMISSION. 
where no special damping is introduced, the viscous friction is largely due to the resist- 
ance of the surrounding air, and if this does not move with the support, it will not be 
proportional to the relative velocity, but to some complicated function of the relative 
and absolute velocities. If the instrument is in a closed room, and the earthquake 
motion not fast, the air will, to some extent, move with the support, and as the damping, 
when special devices are not used, is extremely small, we make no material error in putting 
it proportional to the relative velocity. 
6 in the equation refers to the relative angular displacement of the pendulum; if 
we prefer to deal directly with the relative displacement of the marking point, we can 
proceed as follows: let / be the length of the long arm of the last or marking lever, and 
6 its angular displacement; multiply the equation (25) throughout by nl, where n= 
nnn,- ++. If a is the linear relative displacement of the marking point a=l@= 
nlé; making these substitutes in the equation it becomes 
Yale apt baka i +a) ¥p'= i 
where p'=nlp,. a and its derivatives in this equation will have positive or negative 
signs, according as the number of multiplying levers is odd or even; this will be evident 
if we suppose the support to have an acceleration in the positive direction; the pendulum 
will be left behind; and the long end of the first lever will move in the positive direction, 
that of the second lever in the negative direction, etc. These are the equations of rela- 
tive motion of the pendulum and of the marking point, and they are the only equations 
we have from which to deduce the movement of the support. It will be seen that, for 
the horizontal pendulum with the origin at the CG, and to the degree of approxima- 
tion used, the only displacements of the support which enter are the linear accelera- 
tion parallel to the axis of x, and the rotation around the axis of y; but as both enter 
the equation we are not able to determine the value of either separately. (See, however, 
page 188.) 
For the Milne instrument direct photographic registration is used; if J, is the length 
of the beam from the axis of rotation to the slit for the recording light, then a= —1,0, 
since 1,@ and a are positive in opposite directions; and in equation (26) —nl must be 
replaced by J,. For the von Rebeur-Paschwitz form the optical method of registration 
is used; if D is the distance from the mirror on the pendulum to the recording paper, 
then a= —2Dé@ and nl must be replaced by — 2D. 
We see from the equation that 2 pendulums which have the same values of «, L, 7, and 
p, have identical equations, and their movements for the same disturbance would be 
identical, although they might differ very much in mass and in form; and vice versa, 
in order that 2 pendulums should have identical motions for the same disturbance it 
is necessary that the constants above should have the same values for the 2 pendulums. 
This makes it perfectly clear why 2 dissimilar pendulums give such different records of 
the same disturbance; indeed 2 pendulums made as nearly alike as possible give dis- 
similar records if they have different values of 7, 2.e. different periods; or even if they 
have different values of p’. This was pointed out in 1899 by Dr. O. Hecker.! Two 
horizontal pendulums of the von Rebeur-Paschwitz type made as nearly alike as possible, 
mounted side by side, and having the same period of vibration, gave very different records 
of the same earthquake. The difference was found to be due to differences in the fric- 
tion at the supporting points. Alterations were made until the friction was the same 
in the two instruments as shown by the similarity of the dying-out curves of free vibra- 
tions. After that the two instruments gave similar records of a disturbance. 

1 Zeit. fir Instrumentenkunde, 1899, p. 266. 
