or on the successive gradients, as now altered, shall have less of cuttings and enbankments than in the preceding case. In this way it is advisable to change the positions of the gradients till the minimum of cuttings and embankments seems evidently to be attained, due regard being had to the limit in the ascent or descent of the gradients, which, as before observed, ought not to exceed 1 in 100, also keeping in view, at the same time, the proper height for bridges to cross rivers, &c., in the meantime the difficulty of making the excavations being supposed throughout the length of the line.

Great diversity of opinion prevails among engineers as to the propriety of making the gradients subservient to the economical construction of railways; hence we find, in many cases, a wide departure from the rule, as applied in the earlier days of railways. Some of our most eminent engineers of the present time are in the habit of laying out what are termed " severe gradients," so that inclinations of 1 in 80, and even as far as 1 in 60 are frequently found in the sections of a great number of the new and branch lines, which of course is a question of expediency, between present saving in construction and the future cost of working the lines.

In laying out the gradients, it is desirable to affect as little as possible the existing levels of public roads; which, if practicable, should be crossed either on the level or 20 feet above or below them; if impracticable, the road must be raised or sunk to meet the level of the rails, as the case may require; and sunk to gain the depth of 20 feet, in case of its passing under the railway, the inclination of its approaches to the railway being made 1 in 30 in turnpike, and 1 in 20 in other roads; also, if practicable, all stations should be placed at the top of two gradients descending both ways; as such gradients both serve to check the speed of a train, when approaching such a point, and assist it to regain its speed when leaving.

The height of the gradient over or depth under the surface of any turnpike or public carriage road, existing railway, river, or canal, must be marked in figures on the section at each crossing thereof, as well as the height and span of the arch, or arches, forming the viaducts, by which the crossing is intended to be effected.

RULE FOR FINDING THE RATE OF INCLINATION OF A GRADIENT.

Divide the horizontal length of the gradient in feet, by the difference of the heights of the gradient at its extremities, above the datum line, and the quotient is the horizontal to a rise or

fall of one foot, which is called the rate of inclination of the gradient.

EXAMPLE.

In Plate III. the second gradient at its commencement, is 261-35 feet, and at its termination it is 269.20 feet above the datum line; thus giving a rise of 269.20 — 261·35 = 7·85 feet, the horizontal length of the gradient is 1095.50 — 1064 = 31.50 chains = 2079 feet, whence 2079 ÷ 7.85 = 264, (fractions being omitted) or a rise of 1 in 264, or of 20 feet per mile, as shewn on the section in the plate referred to.

TUNNELS.

When the excavations reach the depth of 60 feet, and continue at that, or a greater, depth for a considerable distance, the most economical method of proceeding with the work is to make a subterraneous passage called a tunnel, through these deep parts; for it would be next to impossible in many such cases to cut the ground open to the surface. Tunnels, on railways of the narrow gauge, are usually cut to the width and depth of 25 or 30 feet, and on railways of the broad gauge they are proportionably larger. The width and depth of the tunnel are less, if the material to be cut be hard rock. A tunnel A B C D, on the gradient NO is shewn in Plate III.; its length AB is 7 chains cr 154 yards, and its height AC 25 feet. All tunnels must slope to one or both of their extremities for the purpose of drainage; and it will be seen that the tunnel, here referred to, slopes to the end A. In laying out the gradients the diminished quantity of cuttings, where there are tunnels, must be taken into account. This subject shall be resumed hereafter.

LEVELLING WITH THE THEODOLITE.

The use of the theodolite is sometimes necessary in levelling operations, especially when these operations are required to be conducted over very high and rapidly rising ground, or over steep and almost perpendicular rocks, where the ordinary levelling instrument cannot be fixed. Select a convenient place to fix the theodolite, where the general inclination of the surface of the country changes, without regarding its minor inequalities; then set the instrument level by means of the parallel plate screws, and direct an assistant to go forward with a staff, having a vane or flag fixed to it, of the same height from the ground as the centre of the axis of the telescope of the theodolite. Having gone to the station required, the assistant must hold the vane staff upright, while the observer measures the vertical angle, which an imaginary line, connect

ing the instrument and staff, makes with the horizon. The instrument and staff should then change places, or, to save time, another staff should take the place of the instrument, the instrument being removed to the former staff, and from thence the angle should again be taken to the second staff; the mean of which two angles may be considered the correct angle. This precaution is necessary on account of the variableness of the refraction, and more especially so where the points of observation are at a great distance, and one much higher than the other. The distance on the slope must be measured in the mean time, which, with the mean angle, constitute the hypothenuse and angle at the base of a right angled triangle, in which the base is the horizontal distance between the two stations, and the perpendicular their difference of level, both of which may be readily found by trigonometry, or by laying down the triangle and measuring the parts in question.

In this manner, by considering the surface of every principal undulation as the hypothenuse of a right angled triangle, the operation of levelling may be carried on with great rapidity; but it must be remarked, without pretensions to strict accuracy, -in fact, in this particular the use of the spirit level can never be superseded.

LEVELLING BY THE BAROMETER.

The method of finding the difference of levels for railway purposes by the barometer, though frequently recommended, will be found to fail in point of accuracy, on account of the sudden changes in the pressure of the atmosphere, on which depend the indications of this instrument, since 90 feet elevation correspond to only one-tenth of an inch of the mercurial column, which difference has frequently been noticed at the same place, in a very short space of time, the weather at the same time being apparently settled. This method, therefore, can never be relied upon further than as a rough approximation.

CHAPTER II.

THE METHOD OF LAYING OUT RAILWAY CURVES, TURNOUTS, CROSSINGS, ETC.

ON RAILWAY CURVES IN GENERAL.

THE natural unevenness of the earth's surface renders the use of curves in railways absolutely necessary, in order that the nearest practicable level may be secured, by avoiding mountains, crags, and other minor elevations, by winding round their bases by means of curves; which are also equally required to avoid various other natural and artificial obstructions, as rivers, lakes, sea-coasts, swamps, &c.; also towns, parks, pleasure-grounds, &c. Thus a great saving is effected in the cost of construction, and in the severance of valuable property, which would be otherwise required. Besides, winding railways are frequently required, in order that they may embrace in their routes important towns, harbours, mineral districts, &c., or make junctions with other railways.

In railway practice the curve adopted is always an arc of a circle; and sometimes two, three, or more consecutive arcs of circles of different radii, having a common tangent or tangents at their point or points of junction, as in the compound curve. Frequently the railway curve is composed of two or more circular arcs, having their convexities turned in different directions, with a common tangent or tangents at their point or points of junction; a curve thus composed is called the serpentine or S curve. curve. The straight portions of the railway are always first laid out, and are, in all cases, tangents to the curves at their termini.

It has been found in practice that at least four different methods of laying out railway curves on the ground are requisite, on account of obstructions on the ground, such as buildings, cliffs, woods, rivers, &c., situated either on the convex or concave side of the curve, or on both sides, or on the curve itself; also on account of pits, bogs, swamps, &c., which either wholly or partly prevent the use of the surveying chain.

Railway curve-rulers are a series of aics of circles of various radii, usually from 3 to 60 inches, and are used for projecting railway curves on parliamentary maps, &c.; and to determine the radii of curves already projected. The radius of

each curve-ruler is marked upon it in inches; and when a curve-ruler is applied to a railway map, the scale of which is 5 chains to an inch, the radius must be multiplied by 5 to obtain the true radius of the curve: thus, if the radius of the curve-ruler be 16 inches, then 16 x 5 = 80 chains one mile, which is the radius of the curve; and so on for maps of other scales.

The limit of the radii of railway curves.-By the Standing Orders of Parliament, a minimum limit of one mile, or 80 chains, was formerly assigned to the radii of railway curves; because, in curves of less radii, a railway-train of great velocity has a tendency to run off the line on the convex side of the curve. This limitation is now very frequently dispensed with, by giving a proper super-elevation to the exterior rail of the curve to counterbalance the centrifugal force. (See formula for this purpose at p. 210.)

The following propositions relating to the circle are derived from geometry, and will be found useful in their application to railway curves.

1. A tangent to a circle is perpendicular to the radius drawn to the tangent point. Thus the tangent A C is perpendicular to the radius A M.

2. Two tangents drawn to a circle from any point are equal, and if a chord be drawn between the two tangents' points, the angles formed by

K

the tangents with the chord are equal. Thus A C=B C, and the angle B A C = AB C.

3. An acute angle between a tangent and a chord is equal to half the central angle subtended by the same chord. Thus CABA M B or A M C.

4. An acute angle having its vertex in the circumference of a circle, and subtended by a chord, is equal to _half_the central angle subtended by the same chord. Thus D A E = DME.

5. Equal chords subtend equal angles at the centre of a circle, and also at the circumference, if the angles are contained in similar segments. The chords A D, DE, E F,