CATENARY

, a curve line which a chain, cord, or such like, forms itself into, by hanging freely from two points of suspension, whether these be in the same horizontal line or not; as the curve ACB, formed by a heavy flexible line suspended by any two points A and B.

The nature of this curve was sought after by Galileo, who thought it was the same with the parabola; but though Jungius detected this mistake, its true nature was not discovered till the year 1691, in consequence of M. John Bernoulli having published it as a problem in the Acta Eruditorum, to the mathematicians of Europe. In 1697 Dr. David Gregory published an investigation of the properties before discovered by Bernoulli and Leibnitz; in which he pretends that an inverted catenary is the best figure for an arch of a bridge &c. See Philos. Trans. abr. vol. 1. pa. 39; also Bernoulli Opera, vol. 1. pa. 48, and vol. 3. pa. 491; and Cotes's Harmon. Mensur. pa. 108.

The catenary is a curve of the mechanical kind, and cannot be expressed by a sinite algebraical equation, in simple terms of its absciss and ordinate; but is easily expressed by means of fluxions; thus if AQ be its axis perpendicular to the horizon, and PQ an ordinate parallel to the same, or perp. to AQ; also pq another ordinate indefinitely near the former, and po parallel to AQ; then, a being some given or constant quantity, the fundamental property of the curve is this, viz, Po : op :: AP : a, or x^. : y^. :: z : a, that is, the fluxion of the axis, is to the fluxion of the ordinate, as the length of the curve is to the given quantity a; where x = AQ, y = PQ, and z = AP. This, and the other properties of the curve, will easily appear from the following considerations: First, supposing the curve hung up by its two points B and C against a perpendicular or upright wall: then, every lower part of the curve being kept in its position by the tension of that which is immediately above it, the lower parts of the curve will retain the same position unvaried, by whatever points it is sus- | pended above; thus, if it were fixed to the wall by the point F, or G, &c, the whole curve CAB will remain just as it was; for the tensions at F and G have the same effect upon the other parts of the curve as when it is fixed by those points: and hence it follows that the tension of the curve at the point A, in the horizontal direction, is a constant quantity, whether the two legs or branches of the curve, on both sides of it, be longer or shorter: which constant tension at A let be denoted by the quantity a.

Now because any portion of the curve, as AP, is kept in its position by three forces, viz, the tensions at its extremities A and P, and its own weight, of which the tension at A acts in the direction AH or po, and the tension at P acts in the direction Pp, and the wt. of the line acts in the perpendicular direction oP; that is, the three forces which retain the curve AP in its position, act in the directions of the sides of the elementary triangle opP; but, by the principles of mechanics, any three forces, keeping a body in equilibrio, are proportional to the three sides of a triangle drawn in the directions in which those forces act; therefore it follows that the forces keeping AP in its position, viz, the tension at A, the tension at P, and the wt. of AP, are respectively as op, pP, and oP, that is, as y^., z^., and x^.. But the tension at A is the constant quantity a, and the wt. of the uniform curve AP may be expounded by its length z; therefore it follows that x^. : y^. :: z : a; which was to be proved.

Also from this last proportion, by proper analogy, or similar combinations of the terms, there arises this other property, ; and the fluents of these give . But at the vertex of the curve, where x = o, and z = o, this becomes ; and therefore by correction the true equation of the fluents is : and hence also . Any of which is the equation of the curve in terms of the arch and its absciss; in which it appears that a + x is the hypothenuse of a right-angled triangle whose two legs are a and z. So that, if in QA and HA produced, there be taken AD = a, and AE = the curve z or AP; then will the hypothenuse DE be = a + x or DQ. And hence, any two of these three, a, x, z, being given, the third is given also.

Again, from the first simple property, viz, x^. : y^. :: z : a, or ax^. = zy, by substituting the value of z above found, it becomes ; and the fluent of this equation is y = 2a X hyp. log. of . But at the vertex of the curve, where x = 0 and y = 0, this becomes 0 = 2a X hyp. log. of √2a; therefore the correct equation of the fluents is y=2a X hyp. log. of ; an equation to the curve also, in terms of x and y, but not in simple algebraic terms. This last equation however may be brought to much simpler terms in different ways; as first by squaring the logarithmic quantity and dividing its coef. by 2, then y = a X hyp. log. of hyp. log. ; and 2d by multiplying both numerator and denominator by , then squaring the product, and dividing the coef. by 2, which gives y = a X hyp. log. hyp. log. hyp. log. .