CATENARY

(Lat. catena, a chain.) ITie curve into which a chord or flexible chain of uniform density and thickne forms itself when suspended or allowed to hang freely from two points. This curve was first noticed by Galileo, who proposed it as the proper figure for an arch of equilibrium; but he imagined it to be the same as the parabola. Its true nature was first demonstrated by James Bernoulli, and its various properties soon after pointed out by John Bernoulli, Huygens, and Leibnitz. It is interesting on account of the light it throws on the theory of arches, and also by reason of its application to the construction of suspension bridges. The equation of the catenary maybe found as follows: let A and B be the points of suspension, and C the lowest point in the curve. Draw C D perpendicular to the horizon, and through any two points P and o very near each otner, draw P D and p d perpendicular to C D; draw also P r parallel to C D, and let PS be the tangent at P. Now, the chain having a umed this form, the equilibrium would not be disturbed by supposing the part CP to become rigid. In this case it would be kept at rest by three forces; namely, the tension at its two extremities, and its own weight. The tension at C is exerted in the direction C E or j» r, that at P in the direction P S or Yp, and the weight in the vertical Vr. These three forces are therefore, by the principles of mechanics, proportional to pr,p P, and Vr; that [s. 214]