THEORY OF THE SEISMOGRAPH. 183 
We suppose that the pivots of the indicator lie on the positive sides of the axes of x and y 
respectively. These equations refer to horizontal indicators with vertical axes of rota- 
tion; the primes and seconds refer to the two indicators. If, as in the Vicentini pendu- 
lum, the first multiplying lever is vertical; then J,.,’=J,,)'’; 0,/ becomes 6,'; and 6,'’ 
becomes @,'’; with these changes equations (119) still hold. Assume that f,=/, and 
f, =f; write 6,'= —n'0,; 0,''= —n'O,, where n’=1,/1,' and n'’=1,/1,"; 1,' and 1, 
are the lengths of the short arms of the indicators. Remembering that f,= —/f,', and 
f,= —f,'', and substituting in equation (118) the values of f, and f, from equation (119), 
we get 
(120) 
26 d? d’6 a 
(Ta) + n'*Zsy') aa =—M!l ‘2 +9 (o, + 4) (La + 2'?Isy'’) ce = Ml = — gq (a, + 42) 
The second equation becomes identical with equation (23) of the horizontal pendulum 
if we replace d?0,/di* by d?6/dt’, and 62 by 78,, and shows that the actions of the two 
types of instruments are the same, but that, other terms in the equations being equal, 
the force of restitution of the horizontal pendulum is only 7 times as great as that of the 
vertical pendulum. The first equation differs only in that d’n/di? has a negative sign; 
this arises from the fact that a positive acceleration of 7 causes a negative acceleration 
of 6,, whereas a positive acceleration of € causes a positive acceleration of 6,; which is 
also true of the horizontal pendulum pointing in the positive direction of y. This differ- 
ence causes no confusion in practice. On introducing terms for viscous damping and 
solid friction, we obtain equa- 
tions exactly like (25) and on x / 
passing to the recording point 
we get equations like (26). 
Therefore all that has been 
developed regarding the hori- 
zontal pendulum — the meth- 
ods of determining the con- 
stants, the magnifying power 
for linear displacements and 
tilts, and the interpretation of 
the record —applies equally 
well to the simple vertical 
pendulum, if we replace 7 by 1. 

THE INVERTED PENDULUM. 

The inverted pendulum 
consists of a mass balanced on 
a point so that its CG is verti- 
cally over the point. This 
position is rendered stable 
either by springs or by a 
second pendulum hanging im- 
mediately above, the two 
being so connected that the af. he We 
points of contact suffer equal 
displacements, and their weights and lengths being so adjusted that the total force 
arising from a displacement tends to bring the system back to its original position. 


