he 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 perpendicularly, 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 IPp depends entirely on the nature of the curve BIG, hut 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 PQ 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 Qp : : It : cos PQp, 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 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 inclination of GO° 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 on a vertical plane perpendicular to the road, the apparent distance Qp is to the actual distance Qp of figure 9 as the sine of the inclination is to radius, wherefore, the apparent breadth Qp on this projection is proportional to 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 conclusions 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 perpendiculars across the successive intervals; but however numerous he may make these sections, he can thus only effect an approximation to the true 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 outline 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 bei