ARCH 331 exposed face of the stone must be right-angled. Now, the projection upon a horizontal surface of a right angle placed obliquely is not necessarily right ; in this case it cannot be right, and therefore the course of a line of joints repre sented in plan must bend away from being perpendicular to the side wall towards being parallel to the line of the abutment. Thus a continuous course of joints beginning at I must be shown in plan by some curved line such as IPp. In many of the skewed bridges actually built, the out line of the arch is divided into equal parts, as seen on the ends of the vault ; the curved joint-lines IPp thus become portions of screws drawn on an oblique cylinder, and, although the arch-stone at the crown be rectangular, those on the slope cease to be so. The bearing surface is thus inclined to the direction of the pressure, and the tendency is to thrust out the arch-stones at the acute corners F and G. The fault is exactly the same as if, in ordinary building, the mason were to bed the stones off the level. The consequence is that skewed stone-bridges have not given satisfaction, the fault being attributed to the principle of the skew, whereas it should have been assigned to the unskilfulness of the design. Let figure 9 be an elevation projected on a vertical plane parallel to AB, EIG, FSH, being the outlines of the ends Fig. 9. of the arch, and the sections taken at equal intervals along the crown line being also shown ; then, since the projection of a right angle upon a plane parallel to one of its sides is always right, the joint at E, as seen on this elevation, must be perpendicular to the curve at R, and thus the curve IPp, representing one of the joint-courses, must cross each of the vertical sections perpendicularly. In this way each of the four-sided curvilinear spaces into which this elevation is divided must be right-angled at its four corners. This law is general, and enables us to determine the details of any proposed oblique arch. If we draw, as in figure 9, the end elevation of the vault as intersected by numerous parallel planes, and lead a curved line crossing all these intersections perpendicu larly, we obtain the end elevation of one of the joint- lines, and are able from it to prepare any other of its projections. The form and character of this end elevation TPp depends entirely on the nature of the curve EIG, but is the same whatever may be the angle of the skew. In order to examine its general character, let us take in the crown line two closely contiguous points I, K, and from these lead the joint-lines IP, KQ, of equal length, then the straight line FQ is equal and parallel to IK, on any of the projections. If in the end elevation, figure 9, we continue the joint IP to meet the vertical section OQ in p, we may regard PQp as a small rectilineal triangle, right angled at p, while PQp is the inclination to the horizon. Now, PQ : Q/> : : II : cos PQ/j, while PQ is equal to KI, the breadth of the arch-stone at the crown, wherefore the breadth of the course at the crown is to the breadth of the same course at any other place as radius is to the cosine of the inclination there. Hence it follows, as is shown in the end elevation, figure 10, that the arch-stones gradually diminish in breadth from the crown downwards, being halved in breadth at au inclination of 60. At a greater inclination they become S Fig. 10. still narrower, and an infinity of them would be needed to reach the abutment of a semicircular or semi-elliptic arch, because the cosine of the inclination there is zero. In no properly-built skewed bridge can the arch-stones show equal divisions ; and it is impossible to continue the arch to the complete half circle or half ellipse. Passing from the end elevation, figure 9, to the plan, figure 8, we observe that Qp on the plan is less than the actual Qp of the elevation in the ratio of the cosine of the inclination to radius, and, therefore, on the plan, the breadth at the crown is to the apparent breadth of the course at any other place as the square of the radius is to the square of the cosine of the inclination there ; so that, at the inclina tion of 60 the apparent breadth will be quarter of that at the crown. Again, in figure 11, which is the side elevation of the vault, or its projection s N , on a vertical plane per pendicular to the road, the apparent distance Qp is to the actual dis tance Qp of figure 9 as the sine of the in clination is to radius, wherefore, the apparent breadth Q/> on this pro jection is proportional to Fig. 11. the product of the sine by the cosine of the inclination, that is, to half the sine of twice the inclination. The width on this projection is therefore greatest at an inclination of 45, being there just one-half of the actual breadth at the crown of the arch. This reasoning is founded on the supposition that the distance IK is excessively small, and the resulting con clusions are strictly true only of an infinitely narrow course of arch-stones; they are, indeed, differential equations which must be integrated in order to be applied to actual practice. Thus we have seen that the curved line IP, figure 9, crosses the section NP perpendicularly at P, but then it does not continue in this direction for any perceptible distance. The draughtsman may attempt to trace it by making the sections very numerous, and by drawing perpen diculars across the successive intervals; but however nume rous he may make these sections, he can thus only effect an approximation to the tme curve. We must integrate, that is, we must obtain the aggregate of an infinite number of infinitely small portions in order to reach an absolutely true result. These conclusions hold good whatever may be the out line of the arch. The most common, and therefore the most interesting case, is when the longitudinal section is circular, the cross section taken perpendicularly to the abutment being then an ellipse with its shorter diameter placed horizontally, the vault being an oblique cylinder. Figure 9 is actually drawn for the circular arch. If then
O be the centre of the circular arc NP, the curve IP must