MECHANICS. 33 
an individual place: if the pendulum be carried to some other place on the 
earth’s surface, where the intensity of gravitation is different, the duration 
of its oscillations will be changed. 
The preceding laws apply only to the mathematical pendulum, and as 
these cannot actually exist, our investigations must have reference to the 
compound pendulum. Suppose in some point of the line AB, a molecule, m, 
and in B the molecule z, then m, being nearer to the point of suspension, 
will make shorter vibrations than n, and will consequently accelerate its 
motion, while 2 will retard the motion of m; oscillations will therefore 
result, such as would be produced by a simple pendulum shorter than AB 
and longer than Am. In every material pendulum, therefore, there must be 
a point whose motion is neither accelerated nor retarded by the rest of the 
mass, and which will consequently oscillate in the same manner as a simple 
pendulum whose length is equal to the distance of this point from the point 
of suspension. This point is called the centre of oscillaticn of the pendulum, 
and when mention is made of the length of a pendulum, by it is always to be 
understood the distance from the point of suspension to the centre of oscilla- 
tion. In very long pendulums composed of very thin threads and very heavy 
balls, the centre of oscillation lies at an inappreciable distance below the 
centre of gravity of the ball attached ; this centre of gravity, therefore, may 
without material error be considered as the centre of oscillation. 
From the preceding considerations it follows that from observation of the 
oscillations of one pendulum, it becomes possible to determine the length of 
another which shall vibrate exact seconds. Borda used a pendulum which 
was exactly twelve Paris feet in length. and made 1876 oscillations in an 
hour. Now, asa seconds’ pendulum must make 3600 oscillations in the same 
time, and the lengths of the pendulums must be as the squares of the times 
144.1876" - 
36007 
= 39.14 Paris inches; more accurately, in English inches, 39.12851. The 
length of the pendulum vibrating seconds at New York is 39.10153 inches. 
If a pendulum could be so constructed as to accomplish its oscillations in 
the are of a cycloid instead of a circle, the length of the pendulum being 
equal to the diameter of the generating circle, all its oscillations would be 
perfectly isochronous; the cycloid possessing the property that great and 
small arcs are traversed in equal times. Huyghens, who probably first 
applied the pendulum to the clock, endeavored to make the pendulum vibrate 
between cycloidal plates or cheeks, so that the thread or spring supplying 
the place of the rod of the pendulum, would be obliged to bend along these 
cheeks; the ball moving, therefore, in a cycloidal curve, and describing 
isochronous oscillations. Nevertheless, the arrangement of these cycloidal 
plates is attended with great difficulties, and for this reason it is generally 
the custom to employ circular pendulums of small amplitude, which have the 
same advantages as the cycloidal, and are of much more easy construction. 
Circular or centrifugal pendulums are those in which the oscillations, 
instead of being performed backwards and forwards in the same vertical 
of oscillation, it follows that 3600? : 18767 ::144:x; therefore z = 
207 
