made consistent with the laws of equilibrium by merely dressing the inner ends of the arch-stones at the appropriate angles. Hence an entirely new field of inquiry, in which we may find the complete solution of the general problem: - " The intrados and extrados of an arch being both prescribed, to arrange the parts consistently with the laws of equilibrium:' Let PQ represent the inner end of one of the arch-stones, the part Qq being vertical, and Pq being sloped at some angle which is to be found ; put t for the tangent of the inclination of the joint P to the vertical, for that of Pq to the horizontal line, then the horizontal strain at P is - I°' while the corresponding strain at Q is w -F&o - ' and if Pq were horizontal these would be alike ; t but the obliquity of Pq causes the load Sw which is placed on it to generate a horizontal pressure 08w, wherefore - t t+St =08w, whence w St 1 Or Ot w St -1.
= - - - - t2 t Sw t Now, when the forms of the intrados and extrados are both given, the values of w, t, 8w, 8t, are thence deducible, so that the value of B may always be computed by help of differentiations only ; excepting, indeed, that integrations may be needed for determining the value of w, which is the area included between the two curves.
In this very simple investigation we have the complete solution of the principal problem in bridge-building. The data needed for determining the shape of the inner end of the arch-stone are already in the hands of the architect, who must know, from his plans, the weight of each part and the inclination of each joint ; so that, with a very small addition to the labour of his calculations, he is enabled to put the structure completely in equilibrium, even on the supposition of there being no cohesion and no friction ; that is to say, he is enabled to obtain the greatest stability of which a structure having the prescribed outlines is susceptible. Even although he may not care to have the stones actually cut to the computed shape, and may regard their usual roughness and the cement as enough, he may judge, by help of the above formula, of the practicability of his design ; for if at any place the value of 08t come out with the wrong sign, that is, if w.8t be less than t.8w, the building is unstable, whereas if w.St be greater than t.8w everywhere, the design, as far as these details go, is a safe one.
In every possible arrangement of the details, the horizontal thrust at the crown of the arch is transmitted to and resisted by the ultimate abutments. The only effect, in this respect, of varieties in the form of construction is to vary the manner of the distribution of that strain among the horizontal courses. Hence one great and essential element of security, - the first thing, indeed, to be seen to, is that the ground at the ends of the proposed bridge be able to resist this out-thrust. Another, and not less important one is, that the arch-stones be able to withstand the strains upon them. In this respect much depends on the workmanship ; it is all important that the stones touch throughout their whole surfaces ; if these surfaces be uneven the stones must necessarily be subjected to transverse strains, and so be liable to fracture. The practice, too common among house-masons, of cheaply obtaining an external appearance of exactitude, by confining their attention to a chisel-breadth around the outside, is not permissible here, nor should any reliance be placed on the layer of mortar for making up the inequalities.
The limit to the span of an arch depends primarily on the quality of the material of the arch-stones. At the crown of the arch the horizontal thrust is the weight of as much of the masonry as fills a rectangle whose length is equal to R, the radius of curvature, and whose breadth is A, the effective thickness there; now this strain has to be borne by the arch-stones, whose depth we shall denote by d, and therefore these stones must be subjected, as it were, to the direct pressure of a vertical column whose height is - d.
This column must be much shorter than that which the