18 J. B. CHERRIMAN : THE MOTION OF A CHAIN, Erc. 
Also 
dy __ Ya— th 
dx 5 nn 
Hence, if a uniform and inextensible chain is moving in any manner on a fixed plane 
curve, its centre of gravity at any instant is moving parallel to the chord of the chain, and 
the velocity of the centre of gravity is to that of the chain in the ratio of the chord to the. 
length of the chain. 
4. If the curve is a cycloid with axis vertical and vertex upwards; and where, if the 
vertex is origin, s — 8ay; the equation of motion becomes 
dv es af ap 
— —= 2 — = = (2ls,—F 
ml US Cats) 
or 
ds g 
Sear ae 
dt? 4 2!) 
= re s, ifs is the distance of the middle point of 
(a 
the chain from the vertex. Hence the cycloid is tautochronous for a chain, and the time 
of oscillation is the same for all chains under all circumstances on a given cycloid. The 
5 5 = 5 \ à m 6 
tension at an assigned point as in (3) is expressed by a 1, 4, and therefore remains the 
l 
32 a 

same throughout the motion; and it has its greatest value at the middle point, being 
x wt. of chain. 
