164 REPORT OF THE CALIFORNIA EARTHQUAKE COMMISSION. 
. dg =a,—2r; and we see that each successive excursion of the pendulum is diminisht 
by 2r.. When at last the friction stops the motion between the lines (1) and (2), the point 
will cease to vibrate, the friction being 
just enough to hold it in the position 
where it stops. But when the vibration 
becomes very small, the friction no 
longer exerts a constant force, but a 
damping force and a force of restitution, 
and therefore the marking point would 
continue to approach the true medial 
line, being kept from it only by the 
constant friction of the pivots. 
When there is damping, the suc- 
cessive excursions about the displaced 
lines are not equal, but they gradually 
, which we have called e; we have therefore 

—KT/2 
diminish in the ratio e 
SO el Ae te (49) 
a+r +r 
and it is from this series of equations that we must determine « and p’. As the position 
of the medial line may be unknown, we can not measure the a’s, so we must proceed as 
follows: adding numerators and denominators of the equal fractions we get 


oe aye at eD ne 
% he 2 hs pice tm a ae Bee (50) 
Ag+ a,+ 27 A,+27r A,+2r 
where A,=a,+a,, A, = 4, + dg, etc., the A’s are the ranges of the vibrations, that is, the 
distances from a maximum excursion on one side to the next on the other. Subtracting 
the numerators and denominators, the second from the first, the third from the second, 
etc., we find 
A ig ye 
do Ag Wd, eae ee oy 

Solving the first equation (50) for 27 and introducing the value of e from the first equa- 
tion (51), we get 
.— Ay — Aids 
er. fF 
21 (52) 
Equations (51) and (52) enable us to determine the values of ¢ and r from the measure 
of three successive ranges; these equations are suitable when the ranges diminish rapidly 
in value; but when they diminish very slowly, these equations will not yield accurate 
values, and we must deduce others containing ranges which are sufficiently far apart to 
have materially different values. We proceed as follows: add the numerators and the 
denominators of equations (51) and we get 
A, —- i Be A, mas Aa = 

yee es Fe Re Ge 
multiplying n of these fractions together, we get 
oe ae a 
Agi — Anim 
m and n may be any numbers we please; let us take m=n+1, and the formula becomes 
A, — Ans = = eT (54) 
Anyi ae Agni 
