ECT. 2.
39
infinitely along the directrix rnn, this line being indeed anasymptote to the said curve.
The calculation in numbers is also equally easy andobvious. Thus, taking any given angle aog, o b being =.Vob 2 — o a 2 , then r.p = oh = o b x sec. aog, and henceoh = op = s /o i/ 2 + l p* ~ f oa 1 + L P^> which gives a pointh in the curve. And the curve thus constructed gives thevery same as the fig. p. 35, formed on the principles of prop.6, as might be expected.
Examples of other curves, besides the circle, might behere taken, but the above case may suffice, as none.of themare of a nature to be suitable for, or to hold good, in theconstruction of arches, at least for the ordinary purpose ofbridges. Because, that in such arches, the parts do not en-deavour to slide down in the oblique direction of the joints,both on account of the roughness or friction there, andbecause, when the parts are cemented together by the mor-ter, or keyed together by pieces within side, the weights thenall act perpendicular to the horizon, being each fixed to theother parts of the arch, after the manner supposed in the9th and 10th propositions; and according to the examplesto the latter of these, it will therefore be expedient to makesuch calculations as may occur in cases of real practice.
PROP. VIII.
When a cum is kept in equilibria, in a vertical position, byloads or weights bearing on every point of it: then the load orvertical pressure on every point, is directly proportional to theproduct of the curvature at that point, and the square of thesecant of the elevation above the horizon of the tangent to thecurve at the same point, the radius being 1. That is, the load.or vertical pressure on any point c, is directly as the cur-vature at c, and as the square of the secant of the angle ben,made by the tangent be and the horizontal line cii.