The equations (4) and (5) give, by addition, Wherefore, C being a constant, the last equation gives by integration sds y + c = √ (s2 + a3) + But when y = 0, s = 0, and therefore C = a. Substituting this value of C in the preceding equation, and reducing the result by (6), we get finally This is the equation of the common catenary, the lowest point of the curve being the origin, and the horizontal tangent at the lowest point the axis of x. The following is another convenient form of the equation of the common catenary in reference to the same coordinate axes. By subtraction of the equations (4) and (5) we get which is the second equation of the common catenary just alluded to. Properties of the Common Catenary. 134. By (7), Art. 132, the tension t at any point P of the is the tension at any point of the common catenary. Comparing this result with (7), Art. 133, we see that the tension t at any point of the common catenary is Hence, if in fig. 64 the axis of y be produced downwards until the produced part AO is equal to a, then by (2) the tension at any point of the common catenary varies as the distance of that point from a horizontal line drawn through the point 0, which is at a distance a from the lowest point A of the curve. The quantity a, as defined in a preceding paragraph, is the length of a uniform chain having the same transverse section as the suspended chain, and whose weight is equal to the tension at A. The horizontal line drawn through the point O is called the Directrix of the Catenary. From (2) we deduce other properties of the common catenary. The tension is evidently least at the lowest point of the curve, and it is the same for two points in the same horizontal line. It follows also from (2) that if a smooth small pulley were placed at any point P of the curve of the common catenary (the part of the chain above the point P being removed), and if over this pulley a chain attached to the chain of the catenary D D were suspended, the length of the suspended chain being equal to the distance of the point P from the directrix of the catenary, and its density and thickness the same as the chain of the catenary, this suspended chain and the chain of the catenary would be in equilibrium. Hence, also, if a cord of constant thickness and density is suspended over two small pulleys, and is at rest by means of certain lengths hanging over the pulleys, the two ends of the cord will be in the same horizontal line. 135. Developing the equation (7), Art. 133, by the exponential theorem, we get Hence, omitting those terms which involve powers of x higher than the second, we get between x and y the relation which is the equation of a parabola. Consequently, the catenary at its lowest point approximately coincides with a parabola. 136. The equations (7) and (8), Art. 133, of the common catenary, are sometimes given in the following forms: By (1) Art. 135, integration by the condition that when y = 0, s = 0, we obtain Integrating this equation and determining the constant from the condition that when x = 0, y = 0, we get the other form of the equation of the common catenary, viz. x = a a log. {2 a (2). THE CATENARY OF EQUAL STRENGTH. 137. PROP. To determine the equation to the catenary when the chain is of uniform density, but the area of the transverse section is such that the thickness at each point of the chain is proportional to the tension at that point, or the strength of the chain at each point is proportional to the tension at the same. The curve assumed by the chain in this case is called the catenary of equal strength. Let A be the lowest point (fig. 65) of the catenary of equal strength, and P any point in the curve. Let the curve be referred to rectangular coordinate axes, which originate at A, the axis of a being horizontal and that of y vertical. Let t be the tension and w the area of the transverse section of the chain at the point P, B the area of H A T' Fig. 65. T M |