In the article BURGEZ the theory has been given of those arches which stand at right-angles to their abutments, and which were the only arches in use until the requirements of canal and railroad engineering suggested the idea of building bridges obliquely across the streams or roads which they have to span.

In these oblique, or skewed arches, as they are called, the plan of the soffit, instead of being rectangular, as in the ordinary bridge, is rhomboidal. Now, if the joints were made to run parallel to the abutments, the directions of the pressures would not be normal to them, and the stability of the arch would depend on the friction of the surfaces, and on the induration of the mortar.

The first attempts to secure the stability of the skewed arch consisted in forming the joints in helical lines, the pitch of the screw being so taken that the line, as it crossed the crown of the arch, might be perpendicular to the parapet, and a lively disputation was carried on in the pages of the Civil Engineer and Architect's Journal, during the years 1838, 1839, 1840, by the advocates of different methods of carrying this idea into effect. Yet a very slight attention to the subject is enough to convince us that the helical joint can only be perpendicular to the lines of pressure at one point, and that, therefore, it cannot satisfy the condition of stability.

The true theory of the oblique arch was first given, by the writer of the present article, in a series of papers read before the Society of Arts for Scotland, in 1835, 1836, 1838, an abridgment of which was published in their Transactions, in the Edinburgh New Philosophical Journal for April 1840, and which was reprinted in the Civil Engineer and Architect's Journal for July 1840.

Let ABDC (fig. 1) be the plan of a skewed arch; AB, CD, being the abutments; AC, BD, the parapets; and KOML the horizontal projection of one of the joints crossing the crown line ON at O. Then, while the lines of pressure must be contained in planes GF, MN, &c., parallel to the parapets, the joint KOML should cross all these upon the surface of the vault, at right angles.

If another joint line be drawn contiguous to KOML, so that the included space may represent the lower faces of an exceedingly thin course of arch-stones, and if MR be drawn parallel to ON, the intercepted distance MR must clearly be equal to OQ. Draw now, on the surface of the vault, the arcs QP, RS, parallel to the parapets, then it can be shown that these minute arcs ought to be proportional to the cosines of their inclinations to the horizon, or that, since O is on the crown of the arch,

\[ \text{arc QP : arc RS} :: R : \cos \text{-incl at M} \]

Hence, if \( a \) be taken to represent the length of the arc NM, \( v \) the distance ON, and \( i \) the inclination of the line of pressure at M to the horizon, so that MR may be the differential of \( v \), and the arc RS the differential of \( a \), \( s \) also being put for the angle of the skew,

\[ \sin s \cdot \delta v = \delta a :: 1 : \cos i ; \text{ or}, \] \[ \delta a = \sin s \cdot \cos i \cdot \delta v \quad (1.) \]

But the projection of the arc RS upon the horizontal plane is less than RS in the ratio of the cosine of the obliquity; wherefore, if \( u \) represent the horizontal projection of NM, and \( \delta u \) that of RS, we have

\[ \delta u = \sin s \cdot \cos i^2 \cdot \delta v \quad (2.) \]

Let fig. 2 be the side elevation of the same arch, that is, its projection upon one of the parapet planes, then, since the projection of a right angle upon a plane parallel to one of its sides is always a right angle, it follows that the end elevation, KOML, of the joint line must cross the projection of each of the lines of pressure perpendicularly. And hence this very remarkable fact, that the configuration of the projection of a joint upon the plane of the parapet is entirely independent of the angle of the skew, and is deducible from the character of the vault alone.

The equations (1) and (2) are universal; they apply alike to circular, elliptical, parabolic, or catenarian arches; the first giving, on being integrated, the development of the surface of the vaults; the second giving the plan.

In the case of the cylindroid arch, when the lines of pressure are circular, the radius being \( r \), we have \( a = r i \); so that equation (1) becomes

\[ \frac{r}{\delta s} = \sin s \cdot \cos i \cdot \delta e; \quad \text{or,} \]

\[ \sin s \cdot \delta v = r \cdot \sec i \cdot \delta t \]

whence by integration

\[ v \cdot \sin s = r \cdot \nep \log \tan \left( \frac{45^\circ + i}{2} \right) \]

or, using common logarithms,

\[ \log \tan \left( \frac{45^\circ + i}{2} \right) = \frac{M \cdot \sin s}{r} \cdot v \]

(3.)

from which equation the positions and dimensions of the arch-stones may be computed with great ease.

The second equation becomes, in the same case,

\[ \sin s \cdot \delta v = \frac{r^2 \delta u}{r - u^2}, \]

whence by integration,

\[ \sin s \cdot v = r \cdot \nep \log \left( \frac{r + u}{r - u} \right) \]

(4.)

So that the horizontal projection of the joint of a skewed circular arch is midway between two logarithmic curves; and in this respect it bears some analogy to the catenary, which is also midway between two logarithmic curves: to this curve I have given the name of double logarithmic.

The projection of the joint of a circular skewed arch, upon the plane of the parapet, is, evidently, the well-known curve called the tractory.

In order to obtain the configuration of the joints for any other kind of arch, we have only to determine, from the nature of the case, the value of \( \cos i \), in terms of \( a \) or of \( u \), and substitute it in equations (1) and (2), which will then, on being integrated, give the development, and the projection on the horizontal plane.

(e. s.)