point. The two closing distances IQ and jQ should be equal, if the ranging has been very carefully done; and this serves as a check on the operation.

In this example the constant decrement of radius of curvature used is 100 feet, but there is no necessity for adhering to such a decrement, the principle can be applied with any other.

Remarks on Ranging Curves for various objects.

Any one of the methods of ranging curves just explained may become suitable under some circumstances. The centre-lines of railways require the most accurate ranging in these days of sharp curves, steep gradients, and high speed; for such purposes, when simple circular curves are adopted, either the two-theodolite or the onetheodolite and chain systems are generally preferred; but if theodolites are not available at the time, or if very windy or extremely cold weather interferes with their convenient use, the equidistant six-point system is probably the most suitable method for ranging; the checking can then be done by oblique offsets.

The curved portions of centre-lines of canals do not demand so much accuracy in ranging as those of railways, for the reasons that an imperfect curve in this case is not a source of positive danger, and that the regularity of the curve is of less importance than keeping close to the intended level, and is liable to alteration from the flow of water. Intervening pieces of straight between two curves are hence frequently dispensed with on canals, and the arrangement and design of curves are made as simple as possible; the radii are always convenient round numbers, and if the two tangent-points are shifted a little forward or back to suit a curve, it does not

become an important matter. The author's equidistant six-point system is well suited to ranging centre-lines of large canals on the curve; and Baker's tangential system may be used for them when the length of curve is comparatively short, but never in very long curves. For centre-lines of small channels, which rarely have long curves, Baker's method is sufficiently accurate; but in sharp curves, which have to run in at some fixed point dependent on level, it is best to use the chord-system, as it ensures the use of the appropriate curvature, and is not practically confined to the use of the radii in round numbers given in ranging tables.

Curves of main-roads and bye-roads correspond to those of main-canals and channels with regard to systems of ranging.

In bridges and viaducts, etc., as it would involve extra labor to set-out and construct a masonry work actually according to the curve, the following plan is

d.

e

FIGURE 36. CURVES ON BRIDGES.

generally adopted. Let acb be (figure 36) the curve which has to be taken over a road by a railway bridge, and which on earthwork would be the centre line used. At c, the middle of the proposed bridge, take a tangent dce, and draw the chord ab parallel to it; the points a and b being sufficiently beyond the two proposed abutments to enable the chord ACB to be so placed that the points A and B will be also a little beyond the abutments; the chord ACB is between the tangent and the

chord acb and is parallel to both;-ACB is the centre line to be used for the bridge, all dimensions being set-out square from it.'

When a viaduct or aqueduct consisting of several spans occurs on a curve, it is necessary that each of the arches should not be conoidal but cylindrical; in order to effect this, the sides of the piers need not be radiated to the centre of the curve, but may be so arranged that each span will be equal on the centre line on the outside and on the inside-as in figure 37; where AB is the chord of the centre line of one opening; instead of using the radial lines cd, ef for the sides of the piers, they are increased and diminished on inside and outside, so that CAD, EBF at right angles to AB, being taken for the pier edges, the spans CE, AB, DF are all equal, and the arch can be made cylindrical; the same method being used for each span of the viaduct or aqueduct.

FIGURE 37. CURVES ON VIADUCTS.

In earthwork on the curve, the curve-points often have to be reset many times; to avoid this, it is usual to set-out two points at equal distances from the required point, and in a straight line with it, at right angles to the tangent at that point. The required point can then be found at any time by bisecting the distance between these two side points, which should of course be In figures 36 and 37 the curves are made very sharp for convenience in illustration.

clear of the works.

It is usual to set them 100 feet

apart. In this manner each point on the centre line, that is liable to be often disturbed during construction, can be found from the two points radially set-out from this point.

In setting-out curved centre-lines on bridges and viaducts the theodolite-and-chain system is generally to be preferred, as want of space may interfere with other methods.

The curved wing-walls of bridges and abutments can be best set-out on the chord system already explained; the radius and the chord can be obtained from the design, and the ordinates from the chord calculated with the help of the table given; sometimes three or four are sufficient instead of nine; and the points set-out with them at foundation-level, and again at plinth-level. Splaying wing-walls of compound curvature may be set-out on the same principle; each portion being treated separately in accordance with its special curvature as shown on the design; but if the curves are sharp, they may be struck from successive centres pegged on the ground.

Section 4. DETAILS IN CONNECTION WITH

RAILWAY CURVES.

Ac

Resistance to Motion on a Curve.-A portion of this resistance will also occur on a straight level line. cording to Mr. Scott Russell's experiments, the following empirical formulæ represent the amount of the various resistances on a straight level.

Ist. The friction of the wheels and axles; this is constant at all velocities, and amounts to six pounds per ton weight of train.

2nd. The resistance of the air, which is proportional to the surface of the front of the train, and to the square of the velocity, amounts to one-fifth of a pound at one mile per hour for the usual frontage of eighty feet.

3rd. The result of concussions, oscillations, flexures, imbedding of wheels in rail, friction of air against sides, etc., when the permanent way is in fair order, equals one-third of a pound per ton per mile per hour, taking it proportional to the weight of the train and the velocity.

The sum of these will give the total resistance, and is true for passenger trains of from 20 to 64 tons at speeds from 30 to 60 miles per hour. For lower velocities, its results are too large, and for carriages or permanent way in bad repair, the results will be too small.

The total resistance to traction at low speeds on a straight may be taken at 1-280th of the load;

at 12 miles an hour, at 1-224th,

at 60 miles an hour, at 1-45th.

It might have been more correct to estimate the resistance of the air in proportion to the bulk of the train, instead of the frontage, but results do not appear to be so. A head or a side wind would add to the total resistance; the former can be accounted for by adding or subtracting its amount from the velocity of the train. in the foregoing formulæ.

A further portion of the resistance to motion on a curve may be due to the gradient.

Should there be any inclination, the resistance due