284 
Proceedings of Royal Society of Edinburgh. [june 20 , 
tlie system is capable of as many simple oscillations as there are 
attached bodies, and all the movements of which it is susceptible are 
compounds of these simple ones. An imaginary flexible heavy line 
may be regarded as composed of an infinite number of parts, and 
thus for it there is an endless series of simple oscillations, each 
having its own periodic time. The essential feature of our inquiry 
is as to the manner of one of these. 
The character of a simple oscillation may be illustrated thus: — 
Let HO, fig. 1, represent the direction of the plummet, while the 
waved line LIHFECBA (supposed, however, to be almost straight) 
is the form of the chain at some particular instant ; then the motions 
must be such that all the parts of the chain come simultaneously 
into the position LtH/EcBO. 
For this it is requisite that the tendency to redress the position 
of any element, as P p, of the chain must be proportional to the 
distance PQ and to the mass of the element. How the tension of 
the chain at P is (in our restricted case) measured by the weight of 
the part below, — that is by the length PA, which we hold as equal 
to QO ; and that tension decomposed in the direction QP is pro- 
portional to the sine of the inclination at P ; so that if we denote 
OQ by 2 , QP by x , and use Leibnitz’ notation, the tendency of the 
dx 
strain on PA to draw the element outwards is proportional to z -7- • 
Similarly, the tension of the chain PE, which is proportional to 0 q, 
dx' 
causes a pull inwardly, indicated by and thus the ultimate 
determination of the element P p towards the central line is pro- 
portional to the difference — 
and thus the general condition of a simple oscillation is expressed 
by the differential equation of the second order 
in which a is a coefficient constant all along the chain, but chang- 
ing from one simple oscillation to another. 
