THEORY OF THE SEISMOGRAPH. 163 
pendulum when the pointer is marking, in comparison to the very slow dying down when 
the marking point does not touch the smoked paper. The effect of the multiplying levers 
in increasing the influence of the friction can easily be found. Using the same notation 
as on pages 151, 155, we have 
Sila = Sal! fala = Sols! 5 etc. 
where for this particular case, the f’s represent the reaction between the levers brought 
about by the friction ¢, of the marking point only, and the inertia of the levers is not 
considered. 
This gives 
| Al ee | eS er Sa 
EL Sod crs ars -=¢@ The ee onl = pml, (47) 
2 atgtg * 8 8 Oy 
that is, the frictional moment is proportional to the multiplying power of the levers. 
Assuming that the friction adds a damping moment, a moment proportional to the 
displacement, and a constant moment, opposing the motion of the pendulum, we have 
still to determine in our general equation (26) the values of the constants « and p’. If 
in this equation we replace a by a! $ Lp'/gi, it becomes 

@a' 5.da' nid , gi/nl \ 9 
da 9, du _! us a! 48 
RAE Peres a) oe 
the form is unchanged except that the constant term drops out. Therefore the vibra- 
tion of a pendulum, affected by constant friction, has the same period and is otherwise 
the same as that of a pendulum without the friction, except that the vibration no longer 
takes place about the medial line, but about a line displaced from it by an amount Lp'/gi, 
and this displacement is first on one side of the medial line and then on the other. We 
may therefore look upon the force of restitution, not as proportional to a, the displace- 
ment, but to a less Lp'/gi; and the pendulum can remain at rest anywhere between the 
two displaced medial lines. Let us call the distance between the true medial line and 
its displaced position, the ‘frictional displacement of the medial line,” and denote its 
value, Lp'/gi or p'(T',/27)*, by r. It must be determined by experiment. We have 
just seen that the frictional moment exerted on the pendulum is proportional to the mul- 
tiplying power of the levers, therefore the frictional displacement of the point 1, is pro- 
portional to the same quantity; and the frictional displacement of the marking point 
is m? times as great, or proportional to the square of the multiplying power. Suppose 
the frictional displacement of the marking point at /, were 0.01 mm., that at the end of 
one lever multiplying 10 times would be 1 mm.; and at the end of a second similar lever, 
100 mm. We can determine the relation between p’, r and ¢; the frictional force ¢ 
exerted at the marking point equals a force m@ exerted at the point of contact, l,, 
of the pendulum and the first lever, and this exerts a moment ¢ml,, and therefore pro- 
duces an acceleration of the pendulum equal to ¢ml,/[J]; this acceleration is represented 
by p, in equation (25). Hence 
20rVv = dmi,nl — pm; 
f—/(—__\r= nl — u — u (48a 
p= (7) roi To oF 

In figure 43, let a), a,, a,, etc., be the successive excursions measured from the medial 
line; let r be the displacement of the medial line; then if there is no damping and the 
pendulum starts from a displacement a,, that is a,—r from the displaced line, it will 
swing an equal distance to the other side of this line, or a,—r=a,+r; ..a,=a,—2r; 
as it starts back from a, the medial line is suddenly displaced to (2), and a,—r=a,+r; 
