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2. Notice of certain Remarkable Laws connected with the 
Oscillations of Flexible Pendulums : With Illustrative 
Experiments. By Edward Sang, Esq. 
In a previous paper there had been given a general method of 
investigating the motions of elastic systems, when the redressing 
tendencies are proportional to the extents of the disturbance. The 
present paper contains the results of the application of this method 
to the oscillations of flexible pendulums, composed of weights attached 
by their centres of gravity to a thread. These results, though 
strictly applicable only to oscillations of infinitely small extent, may 
yet be held as indicating the general characters of oscillations of 
moderate dimensions. 
The exceedingly complex motions of such a pendulum may always 
be resolved into as many simple oscillations as there are bodies in 
the system ; a simple oscillation being such that, if it subsisted 
alone, all the moving points would pass through their mean posi- 
tions at the same instant, and would also all reach the extreme 
limits of their motion at once. 
The periodic times and the configurations of those simple motions 
can be computed by help of an equation rising to the degree indi- 
cated by the number of the bodies ; and we obtain this very singular 
result, that “ Whatever may he the details of the system , the sum of 
the squares of these 'periodic times is equal to the square of the 
periodic time of a simple pendulum having the entire length of the 
flexible one;” or, in other words, that u If simple pendulums he con- 
structed vibrating in accordance with the simple oscillations of such 
a fexible series , the sum of their lengths is equal to the whole length 
of the fexible line” 
In order to cause any given pendulum to perform one of its simple 
oscillations, we would need to give to each one of its component parts 
a properly regulated impulse : thus we might compute their extreme 
positions ; and, having placed each body properly, let them all go at 
the same instant. Hence it is a matter of great difficulty to exhibit 
a simple oscillation when the system consists of even so few as three 
or four parts. 
When all the bodies are of one weight and uniformly distributed 
along the chord, the configuration of a simple oscillation is given by 
the formula — 
