Chap. XXV.

ON STONE BRIDGES.

1489

axis*, it has this advantage over the preceding form, and is more easily described, but itis attended with an inconvenience common to most of these curves, viz. its disagreeableappearance to the eye: in the parabola the value of the radius of the curve is a minimumto the summit of the curve, and from the position of the summit, the curve does notdiminish progressively from the springing to the highest point of the arch.

It has also been proposed to form flat arches with two arcs of a circle, setting off fromthe springing and united at the summit of the vault by a portion of the catenary. Curvesof this kind have a greater water-way than the ordinary segmental arches, and are certainlyto be preferred on this account. The ellipsis whose curvature diminishes progressivelyfrom the springing to the summit is decidedly the most elegant figure that can be adoptedfor the intrados of flat arches: it may be readily drawn by the following simple method,which is recommended by M. de Prony.

Given half the great arch C A,and the lesser semi-axis C B of theellipsis, describe from the point C,with C A for a radius, a circle; thenmove a square so that one of its sidesnF, passing continually through thefocus F, the summit n of the rightangle shall always follow this circle;the other side nt of the square willbe a tangent to the ellipsis: by tra-cing thus p number of tangents, thecurve may be obtained with thegreatest accuracy, and we shall haveat the same time the direction of the tangents and the normals.

The same method may be applied to the hyperbola : for the parabola, the circle becomesa right line tangent to the summit of the curve. This process possesses the advantage ofnot requiring an area greater than the rectangle circumscribing the curve. M. dc Pronyhas also given a very simple formula for determining the direction of the tangent /T, whenthe point t is given, as well as the position of the point >«» by means of its co-ordinatesA p,mp.

Calling A the majorsemi-axis A C;

B the minor semi-axis

BC;

x and y the co-ordi-nates Ap,mp ;

a the distance A t ;

6 the distance A T ;

it the distance Df;and for conciseness mak-ing.

B

— = Y, we havea

y=: l' /2Ax-;r *-

a — -

Bj

V 2 A x — x.

2 A 2 By

, 2 A , 2 A

n=—3-, k = -.

X*-l +1

Fig. 2710.

These formula?, the calculation of which is very easy, may be used with advantage. Whenthe curve of great arches is formed by the arc of a circle, the direction of the tangents, andthe position of a great number of points, should be fixed by a method analogous to thepreceding.

A very simple process, and not requiring any instrument, may be found useful to theengineer : having the major and minor semi-axis, set off their respective lengths from thesame point on a slip of card, (or a rod, if a large ellipsis be required); then commencingfrom one extremity of the minor semi-axis, as A, turn the slip of card in the direction ofa to A, observing always to keep the point B continually touching the major semi-axis,and C the minor; the card or rod will then assume the positions marked a,b,c, and bykeeping a pencil at a , the curve is drawn at once: a,b,c , will be the joints of an ellipticalarch.