5

CATENARY OF EQUAL STRENGTH.

113

Cor. 1 . If a uniform chain hang freely over any twopoints, the extremities of the chain will lie in the same horizontalline when the chain is in equilibrium.

0

Prop. A chain of variable thickness, but of the samematerial throughout, is suspended from two points : requiredto find the law of the thickness that the tension at differentparts of the chain may vary as the strength of the chain atthose parts.

140. Let S be the length of a uniform chain of which thethickness equals that at the lowest point, and weight equals theweight of the length s of the chain to he suspended.

Let, as before, C be the lowest point (fig. 58.) : CM = a?,MP = y, CP - s : c the length of uniform chain of the thick-ness at C, of which the weight equals the tension at C. Theportion CP when it has assumed its form of equilibrium maybe supposed to become rigid. The forces which retain it inequilibrium are its weight and the tensions at C and P, andthese are parallel to the sides of the triangle MTP;

and ••• PT = \/PM*+ MT 2 ;

a/c 2 + .S' 2

.-. tension at P = -tension at C.

c

But the thickness of the chain at P varies ultimately as the

.-. tension at P =

quantity of material in a given short length Ss of the chain,

dS

since the density is constant: it therefore varies as —. But

ds

by the hypothesis the tension must vary as the thickness of thechain;

dS vW .S’ 2

vW .v 2

varies as

since S and a are ultimately equal;

S + v 7 .? 2

P

(fi).