256 
[ , n / l\ nn — \(l\ 2 nn—ln — 2/l\ 3 
— r-iW + i 4 y -I—- 9-y +&c - 
in which l is the length of one of the links, p that of a simple pen- 
dulum oscillating in the same time, x x the ordinate of the lowest 
body A, and x n that of a body situated n intervals above A. 
When a flexible pendulum consists of two weights A and B, 
attached by their centres of gravity to two threads AB, BC of equal 
lengths, the ratio of the periodic times of its two simple oscillations 
may be exhibited by constructing a right-angled trigon PQX, such 
that the square of PQ, may be proportional to the weight A, the square 
of QR to the weight B ; this being done, the periodic time of the 
slower is to that of the quicker oscillation as QP + PR : QR, 
The truth of this law was exhibited experimentally by making A 
nine and B sixteen ounces : an arrangement which gives the periodic 
times as 2 to 1 ; and also by making A sixteen and B nine, in which 
case the ratio is as 3 to 1. 
When, by augmenting indefinitely the number of the weights, we 
pass from the discrete series to a continuous uniform flexible line, 
the previous equation takes the form 
in which p is the length of the corresponding simple pendulum, z the 
distance upwards from the lowest point A, and x x the horizontal ordi- 
nate of A. These formula differ, the one from the well-known develop- 
( l \ n £. 
1 } , the other from e p in having; for the deno- 
pj 
minators of the successive co-efficient the squares of the natural 
numbers instead of those numbers themselves. 
From this latter equation it follows, that if a uniform flexible 
chain could be made to perform one of its simple oscillations, its con- 
figuration would always have the character of the curve so indicated. 
Starting from the lowest point in a direction which, if continued, 
would cross the vertical axis at the height p , this Gurve crosses and 
re-crosses that axis, each wave being longer and flatter than the pre- 
ceding : the first crossing is at the height px 1*445 7965; the 
second at p x 7*61? 8156; and the third at p x 1 8*721 7517 : hence 
