FIGURE 33. THE SINGLE CHORD SYSTEM. This method may be occasionally useful in small curves of roads, paths, small channels, etc. The chord of the intended whole curve is measured, the angle of deflection observed roughly with a compass, and the suitable radius of curvature calculated. See Formulæ, page 214. C=2R sin y. Set out nine points along the chord at equal intervals, each being one-tenth of the chord; calculate the necessary offsets, which will only be five in number, as they recur in the second half of the curve, and set them out, thus obtaining nine points in the curve, which are sufficient for rough purposes of this description. For example. Let the whole chord be found to be 131 feet, the angle of deflection 17 degrees (minutes are unnecessary), then the radius suitable to the case will In the attached table the five ordinates are given to a radius of 1000 feet, and can be thus reduced to the radius of 224 feet : Ist ordinate=15'96 x 224÷1000= 3.6 feet = 3rd ordinate 36.83 x 224÷1000= 8.2 feet 4th ordinate=4198 x 224+1000= 9'5 feet middle ordinate=43'70 × 224÷1000=108 feet The setting-out is now effected by measuring ten distances along the chord from one tangent point, each distance being a tenth of the chord, that is 131 feet; and these points are marked. At the first of these points set-out the offset 3.6 feet at right angles to the chord, at the second 6'3 feet, at the third 8.2 feet, at the fourth 9.5 feet, at the fifth or middle point 108 feet, at the sixth 9.5 feet, at the seventh 8.2 feet, at the eighth 6'3 feet, at the ninth 36 feet. The extremities of the offsets will be points in the curve. This system, though not a very good one, is preferable to the method of versed sines, which is certainly troublesome in calculation. OFFSETS FROM THE CHORD FOR A RADIUS OF 1000. Checking equidistant points already ranged. When a number of equidistant points have been ranged on a curve, they may be rapidly checked by either of the two following methods: 1. By equal rectangular offsets. Produce the chord between any two points, until a rectangular offset hits off the next point; such rectangular offsets should all be equal, and when the chords are small compared with the radius, the offset is equal to double the ordinate from the tangent of the first point set out. 2. By oblique offsets. Produce the chord between any two points to a distance equal to the chord itself, and mark the point; its distance from the next point on the curve is nearly equal to the square of the chord divided by the radius, but in any case such oblique offsets should be all equal. The bisection of such an oblique offset will give the direction of the tangent from the last point on the curve. These two methods are sometimes adopted for ranging curves, when accuracy is unimportant: but they are much better adapted for checking with precision. Curves of varying Curvature. Although ordinary circular curves, as before described and applied, are the most simple and in general the most convenient for almost all purposes, for roads, canals, or railways; yet in the case of a railway, their suddenness of departure from and return to a piece of straight is an acknowledged defect, and in cases where there is danger of a train leaving the rails, these are most dangerous points. In practice, or, more properly speaking in actual fact, in which theory does not inter fere, the sudden changes of cant are merely eased by the plate-layers on laying the rails, in an empirical way. Two methods have been proposed to remedy this, both substituting other and more complicated curves for the circular one;-the one, by Gravatt, gives a varying curvature, the ordinates of which vary with the sine of a theoretical and abstract angle, of which the abscissa, the half-chord, and are the factors; the resulting curve is π 2 supposed nearly to meet the requirements of the cant, which again must be afterwards adapted to it; the other, by Froude, is far more practical in tendency, having a curvature made to depend on the maximum cant and the length of the radii of the adjoining circular portions of curve; the equation used is that of a cubic parabola. In this latter case, working out the ordinates for each single case would take up time, and apparently the setting-out of the curve would also be complicated and tedious. The Compound Circular Curve. There is, however, no absolute necessity under any circumstances for setting-out curves of varying eurvature by making use of complicated expressions for ordinates, nor would there be any advantage in applying curves of a high order, if the same object can be obtained by simpler means. Now, in actual fact, a real curve is a comparative rarity on a railway, for it generally is nothing but a succession of small pieces of straight rail; cambering rails being still more the exception than the rule; so in practice we can make a curve of varying curvature to consist of a large number of equal portions of circular curve, described with different radii, decreasing from the |