Again, we have as sin. d c 1: sin. r c 1: : tension r c : ten r c X sin. r c / sion d C d cbut as r c is inversely as sine sin. r c / ? c d, therefore tension d c is as sin. red X sin. d c Now, let an unlimited number of weights be hung from the chord, indefinitely near each other, and our polygonal thread becomes a curve ; Figure 4 being in fact the curve of equilibration adapted to the weight which depends from it. The angles ? c d and 1 c d become r' c d and /' c d, which are supple mentary, and have equal sines, wherefore the product of these sines is the square of either. Also, as the sine of r c 1 or r c r' is as the curvature, or reciprocally as the radius of cur vature, we have tension d c, or weight on c, inversely as rad. cure. X sin.' inclination to vertical.
" This tension, in the present case, is usually produced by the gravity of the superincumbent materials, and may be . measured by the area contained between two indefinitely near vertical lines E F, e f, Figure 4; but while the distance E e is constant, the area F e will diminish with the sine of F E e, as E e becomes more upright. Tu countervail this, we must enlarge the depth E F in the same proportion as sine c E F diminishes. And, therefore, we have E F inversely as rad. cur. x sin. s F E e. That is, the height of the superin cumbent mailer must be inversely as the radius of curvature, into the cube of the sine of the inclination of the curve to the vertical.
" Let us proceed to apply the theory to some practical cases.
" If the arch be the segment of a circle, then the radius of curvature is the same throughout, and the height will be in versely as the cube of the sine of inclination to the vertical. And from this we derive the following very simple construc tion, for describing the equilibrating extrados of a circular arch, and which the reader, who has examined this subject, will find much easier than those commonly given.
" Figure 5.—At any point, D, draw the vertical n d, and n F from the centre c; then laying off D a equal to the thick ness at the crown, draw the perpendiculars a b, b c, c d suc cessively, D d is the vertical thickness at D, or d is a point in the extrados.
" For it is evident, that Da:Db::Dbinc: :DC:Dd, because of similar triangles; therefore, D a:ad:: rad. : a n b, or inversely as radius to cube sine a b D. Now u a is the thickness at crown, and D d is therefore the thick ness at D. Figure 6 is constructed in this way, and may serve as a specimen of the equilibrating extrados for a semi. circular arch. By reversing this operation, we may find the
thickness at the crown corresponding to a given thickness at any other point. And here we may observe, that as a, Figure 5, approaches the extremity, n, of the semicircle, the line n d rapidly increases, until, at the point It, it is of an infinite length. But indeed this must evidently be the ease with every arch which springs at right angles with the hori zontal line ; for the thrust of the arch should be resisted by a lateral pressure, and no vertical pressure can act laterally on a vertical line.
" We may also observe, that since the extrados or upper outline descends first on each side of the crown, and then ascends with an infinite arc, there is, for any thickness of the crown, a point on each side where the upper edge of the extrados is on a level with that on the crown. Thus, if D n = 30°, its sine is half the radius. n a is therefore = of D d, so that if vv=na be made of v c, the radius, we have the point d at the same level with v. Between this point, however, and the crown, there is a considerable de pression, which is increased if the crown be made still thin ner. On the other hand, if it be made thicker, the horizon tal line drawn through the crown cuts the extrados much nearer the middle of the arch. It appears, therefore, that the circle is not well adapted for the purposes of a bridge, or a road, where the roadway must necessarily be nearly level; for no part of the extrados of the circular arch will coincide with the horizontal line. There is, indeed, a certain span, with a corresponding thickness at the crown, where the out line differs least from the horizontal ; that is, an arch of about 54 degrees, with a thickness at the crown about 4 of the span. But that is far too great for practical purposes.
" We may, however, extend the construction just given, even to those arches that are formed of portions of circles, differing in curvature. For the equilibrating extrados being first constructed for that portion of the arch in which the crown is, as tar as the vertical line passing through the con tact of the neighbouring curves, the thickness of the crown must be supposed to be enlarged, in proportion to the dimi nution of the radius of curvature, or the contrary, and, with this, proceed as befbre along the succeeding branch of the curve. This will, indeed, cause an unsightly break in the extrados, for which we shall not at present pretend to find any other remedy, than using materials of a different specific gravity.