the point P, r being the radius of the arch, the distance IN or i0 of figure 10 is clearly v sin s, and as the result of the integration, we obtain - v sin s - Nap. log tan (450 + , by help of which equation we can readily determine i when v is known, or v when i is given. The table of Napierian logarithmic tangents being very scarce, it is convenient to convert these into denary or common logarithms. Putting, as is usual, M for the modulus of denary logarithms, that is, for '43429 44819, the above equation becomes - - r . v . sin s = log tan (45° + from which it is quite easy to tabulate the values of i corresponding to equidifferent values of v, because the constant factor - has to be only once computed; i, that is, the number of degrees in the arc NP being thus computed for each of the successive sections of the vault, we have only to divide a tape-line so as to show degrees and minutes of the actual circle in order to be able at once to mark the course of the joints upon the centering of the arch; or better still, instead of the degrees, we may write upon the tape the successive values of NP, and then the commonest workman will Le able to lay off the lines.
The only other kind of skewed arch likely to possess any interest is the elliptic. In right arches the semi•ellipse is sometimes used on account of the grace of its form, but this reason for its adoption disappears in the case of the skew, because then we can only use a portion of the semi-ellipse. The end elevation of a joint in an elliptic skewed arch is a modified form of the tractory, and the general features of the arrangement are analogous to those of the circular arch.
The arch-stone of a common bridge is wedge-shaped, having two flat faces AacC, BbdD, inclined to suit the breadth of the course, but in the skewed bridge the corresponding faces are twisted, Cc not being parallel to Aa, and thus the dressing of them requires both skill and care. The dimensions of the stone and the inclinations of its four edges may easily be computed when its intended position is known, and thus the degree of twist on each of its faces may be ascertained, and the lines may then be marked off on the ends of the stone.
The theory of the skewed arch was given for the first time in the Transactions of the Royal Scottish Society of Arts for 1833 ; from which it was copied into the Civil Engineer and Architect's Journal for July 1840, which see. (For the history and various forms of the arch see Aneni-