THEORY OF THE SEISMOGRAPH. 161 
The use of this formula is the quickest means of calculating the value of «T [amr 
which enters the expression for the magnifying power of the seismograph for Ranconic 
vibrations. 
The vibrations of the pendulum under damping lie between two exponential curves, 
e“ and —e™“ as shown in figure 40. 
There are few instruments free of all solid friction; this enters at the pivots and at 
the marking point. At the pivot it is merely a constant moment tending to stop the 
motion; but it may have a somewhat 
different value for motion in opposite 
direction. At the marking point the 
effect is different; in figure 41, let a be 
the pivot, and b the marking point of 
the indicator; let the recording paper be 
moving to the right with a velocity of 
v’’; let the marking point be moving to 
reduce 6’ with a velocity v’; be and bd, 
as shown in the figure, will indicate the 
movements of the marking point relative 
to the paper, as the result of these move- 
ments respectively ; the resultant relative 
motion will be be, and the frictional force 
which will be directed in the direction opposite to be may be represented by a con- 
stant ¢. Let « be the angle which its direction makes with the direction of motion of 
the paper, and let 6’ be the angular displacement of the lever from the same direction 
(which should be its direction of equilibrium). 
We may divide ¢ into two components, one in the direction of the lever, which is 
resisted by a reaction at the pivot and does not tend to rotate the lever; a second at 



Fig. 41. 
right angles to the lever, which exercises a moment to turn it; to determine this moment 
we must get the component of ¢ in the direction of v’ and multiply it by /’. This effective 
moment is 
Se Se ee v' —v"''sin 6! 
gl! sin (a — 6’) = cae pl am ee Sate TY a (45) 

This can be developed in powers of v'/v'’ (which we will write v,,') or of v'’/v' (or v1’) 
whichever is less than unity, and we get 
gl! (1 — v4!" sin 6") (1 — vy"2/2 + v,"' sin 6’ +++ -) 
or gl! (vy! — sin 6") (1 — vy)!2/2 + vy! sin 6! + - - -) (46) 
If the lever is moving very rapidly in comparison with the paper, v,'’ becomes a small 
quantity, it may be neglected, and the first expression becomes @i’, that is, there is a 
constant moment tending to stop the motion of the pendulum. If the paper is moy- 
ing very rapidly in comparison with the lever, ate is a small quantity, and the second 
expression reduces to l'(v,,!—sin 6’—»,,' sin? 6’!+---); which, when @’ is small, 
M 
