 shall obtain a representation of the curve, which will be nearer to the truth as the intervals are more numerous. The beginning of the curve at A may be made a short are of a circle described from the centre O.
Since the minute differentials thus obtained are proportional to the sides of a triangle whose hypotenuse is D2 - A2, and one of whose sides is D2 - z2, we must have - 8x - 4./ ( (D2 _ A2)2_ (D2_ 2,2)2) 8> , and the integration of this would give the value of x. If we put 0 for the inclination of the curve at any point P, D2 - z2 = (D9 - A2) cos 0, ••• z= (D2_ (D2 _ A 2) cos OP, and taking the differential, Sz = RD?. = .A.2) sin st,(D2 - (D2 - A2) cos CI 80, II. cos p. .•. 8x - %/(D2-21-1 cos p) where 211 is put for its equivalent D2 - A2. The integral of this expression may be obtained by developing the radical in terms arranged according to the powers of cos 0, and then integrating each term separately. The result is a series of terms proceeding by the powers of cos 0, the coefficient of each power being itself an interminate series ; and the rate of convergence is so slow as to make the labour of the calculations very great. Such expressions belong to the class of elliptic functions, for which peculiar methods have been devised. Fortunately the actual calculation is not required in the practice of bridge - building, and therefore we shall only refer the reader to the above-named subject.
If the horizontal thrust and the thickness at the crown of the arch be prescribed, the radius of curvature there must be the same whichever of the two hypotheses be adopted ; now, if we sweep an arch from the centre C with the radius CA, the catenarian curve lies outside of it, while the curve which we have just been considering lies inside. Each of these is compatible with sound principles: the one if the inner ends of the arch-stones be dressed with horizontal facets, the other if the ends be dressed to a continuous curve ; wherefore, between these two limits we may have a vast variety of forms, each of which may be 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