SECT. 2,

OF THE ARCHES.

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of the hanging festoon abcde. By varying the distance ae,of the points of suspension, moving them nearer to, or fartheroff, the chain will take different forms ; then the frame abcdemay be made similar to that form which has the most pleas-ing or convenient shape, found above as a model.

Indeed this principle is very fruitful in its practical conse-quences. It is easy to perceive that it contains the wholetheory of the construction of arches: for each stone of ailarch may be considered as one of the rafters or beams in theforegoing frames, since the whole is sustained by the mereprinciple of equilibration, and the method, in its application,will afford some elegant and simple solutions of the most dif-ficult cases of this important problem; some examples ofwhich will be shown hereafter.

prop. v.

To form mechanically a balanced Festoon arch, on the prin-ciples of the last proposition ; having a given pitch or heightand span, and also a given height and form of wall or roadwayover it.

Let am be the givenor proposed span of thearch, pn its pitch orgreatest height, dk thethickness at the crown,and alknm the given an-terior form of the wall:in order to determine theform of the curve admwhich shall put that wallin equilibrio.

Invert the whole figure alknm, as in the opposite posi-tion hlknu, or construct this latter figure, on the lower sideof am, exactly equal and similar to the proposed upper one;the point d answering to the point D, and the point k to the

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