marked, and a line ruled in the place occupied by it for the first gradient. The second, third, &c., gradients are laid out in a similar manner to the end of the section. See Art. 26 and the two follow

ing notes.

NOTE 1.-The excavations and embankments of a railway are made about 2 feet lower than the level of the rails, thus giving a line parallel thereto, called the balance or formation line; the 2 feet filled up with gravel, to form the road and the beds for the sleepers of the rails.

2. If the position of the first gradient, though favourable in itself, cause the following gradient, or gradients, to be less favourable, with respect to the quantity of cuttings and embankments, it is advisable to alter the position of the first gradient to one less favourable, provided that the compound results of cuttings and embankments on the several successive gradients, as now altered, is more favourable than in the preceding case. In this manner, it is requisite to change the positions of the several gradients repeatedly till the minimum, or least possible quantity of cuttings and embankments, shall be required in the construction of the railway, keeping in view the required limit in the ascent and descent of the several gradients; the difficulty of making the excavations, throughout the whole length of the line, being supposed, at the same time, to be nearly equal. But where the geological character of the country through which the railway passes, differs considerably, presenting for excavation strata varying throughout the length of the line from loose sand to hard rock, and vice versa, the facility or difficulty of making the excavations must be carefully considered in laying out the gradients; larger excavations being advisable where they can be easily made and smaller where with difficulty. The least possible expense will be incurred in the construction of a railway by taking into account all these circumstances.

(25.) If anywhere in the section the excavations reach 60 feet in depth, and afterwards increase rapidly in depth, it is a more economical method of proceeding to make a subterraneous passage, called a tunnel, through the deep part, than to cut the whole open to the surface of the ground, which in many cases would be next to impossible. Tunnels are cut to the width und depth of 30 feet, for railways on the narrow gauge; if on the broad gauge, to the width and depth of 36 and 32 feet respectively, the width and depth being less in both cases, if the material to be cut be hard rock. The diminished quantity of cuttings, where tunnels occur, must be taken into account in laying out the gradients.

(26.) To determine the rate of inclination of a gradient.

AB, in Plate XIII., is the first gradient on the railway section, the cuttings or excavations above it being considered to be equal to the requirements of the embankments below it: at its commencement A, its height AO is 100 feet; and at its termination B, its height BC is 130 feet above the datum line; thus giving a rise of 130-100

=

= 30 feet; the horizontal length oc of the gradient is 65.60 chains -4329-6 feet. There is, therefore, a vertical rise of 30 feet in a horizontal distance of 4329-6 hence 30: 4329.6 : 1:144 32; or, in round numbers, a rise of 1 in 144, which is called the inclination of the gradient AB, and is thus noted on the section, INCLINATION 1 IN 144.

The rule for finding the rate of inclination of a gradient may be thus briefly enunciated :

Multiply the horizontal length of the gradient in chains by 66, and divide the product by the difference of the heights of the gradient at its extremities, above the datum line, and the quotient is the horizontal distance to a rise of 1 foot, or the rate of inclination required.

(27.) The excavations at B being about 60 feet deep, and continuing to increase rapidly, a tunnel BD, to the length of 462 yards, is introduced on the next following gradient; the tunnel having passed beyond the summit, till its depth at D again becomes 60 feet; its height, as shown on the section, is 21 feet, being the height to which the cuttings are reduced by the ballasting below and the arch above. The gradient, of which the tunnel forms a part, is assumed to extend beyond the limits of the plate; its position, therefore, is not determined by the excavations and embankments shown thereon, but in conjunction with those beyond its limits.

(28.) Method of determining the heights of the several roads passed over or under by the railway, and whether they should be raised or lowered to give sufficient height for viaducts or bridges; or to be raised or lowered to be passed on the level of the railway.

When a road is passed over by the railway, the usual height allowed from the surface of the road to that of the rails is 18 or 20 feet; and when the road is passed under by the railway, the height allowed is 18 or 19 feet.

The heights or depths of the roads, rivers, &c., above or below the rails, or gradients, is sometimes found by measuring them carefully by the vertical scale; but they may be found more accurately by the following method:

The distance of the middle of the first road from the commencement of the section is 10 chains, its height above the datum lines is 83-20 feet, the horizontal length oc of the first gradient is 65.60 chains, and the gradient rises 130-100 = 30 feet. There is, therefore, given the length and rise of the gradient to find its rise at any other given point; which may be done by similar triangles; thus

As hor. length of gradient 65.60 ch. : 30 ft.:: 10 ch. the dist. of road: 4.57 ft., the rise of the gradient at a; this added to 100 ft., the height of the gradient at A, gives 104-57 ft., the height of the point a above the datum line; but the height of the road above the same line is 83.20 feet, therefore 104:57 -83.20 21.37 ft. = 21 ft. 4 in. is the height of the rails at a above the road; which, being above 20 ft., shows that the level of the road may remain unaltered, the height of the bridge required for the railway being 18 ft.; thus leaving 3 ft. 4 in. for the thickness of the arch and ballasting, and its span being taken 24 ft., as being sufficient to allow carriages to pass one another on the road below the line.

The distance of the occupation road per Level-Book, is 24-06 ch., and the reduced level is 110 ft., therefore 65.60 ch. : 30 ft.:: 24-06 ch. 10-91 ft.; hence

100+10-91110-91 ft. height of gradient at road

110-00 ft. do. of road

0.91 ft. 11 inches, the height the road must be raised to be passed over on level of rails. In raising or lowering roads for this purpose, a rise or fall of 1 foot in 20 is required.

In the same manner, the road from Westbrook to Hurst is found to be 22 feet above the gradient; the hedge of the brook that forms the boundary of the parishes of Westbrook and Winston, 36 ft. 5 in. below the gradient; and the road next following, 40 ft. 5 in. above it.

SECTION III.

RAILWAY CURVES.

(1.) On the use of curves in railways in general.

The use of curves in railways is absolutely necessary on account of the natural unevenness of the earth's surface, it being desirable to attain the nearest possible practical level by avoiding hills, crags, mountains, &c., and by winding round them by means of curves. Curves are also equally necessary in avoiding other natural and artificial obstructions, in many cases not materially affecting the level of the line, as lakes, swamps, the windings of sea-coasts, and of rivers; also cities, towns, villages, parks, pleasure-grounds, &c. In this manner a great saving is effected in the expense of extensive excavations, embankments, tunnels, viaducts, &c., as well as the expense of the severance of valuable property, which would otherwise be required. Besides, it is frequently desirable to make a

winding railway, in order that it may embrace in its route some important city, town, harbour, &c., or make a junction with another railway. Straight lines in railways are, however, much to be preferred to curves, and are therefore adopted as far as possible ; they are first set out as bases for further operations; and every curve is subordinate to two straight portions of the line, which are tangents to it at its commencement and termination. The curves adopted in practice are always arcs of circles. Sometimes two, three, or more consecutive circular arcs, having a common tangent at their point or points of junction, are joined together, as in the case of the compound curve; and sometimes two circular arcs are connected, having their convexities turned different ways, with a common tangent at their point of junction, as in the case of the serpentine, or S curve.

NOTE.-Other curves beside the circular arc might sometimes with advantage be adopted as the curves of railways; but their construction is attended with difficulty.

(2.) On mechanical railway curves, or curve-rulers.

Mechanical railway curves, sometimes called curve-rulers, are a series of segments of circles made of hard wood, as box or mahogany, or strong paste-board,

with their radii in

inches marked on

them. These curves

16

commonly begin with a radius of 24 inches, and terminate with one of 160 inches, or upwards; the smallest radii usually increased by inch up to 10 inches; then by inches up to 20 inches; afterwards by 2 inches up to 80, including all numbers ending in 5; lastly, by 5 inches, till near the end of the series, when the radii increase by 10 inches.

The annexed figure represents the railway curve-ruler of 16 inches radius. If the scale of plan, to which this curve-ruler is applied be 5 chains to an inch, it will represent a curve or circular arc of 16 × 580 chains = 1 mile radius. If the scale of the plan be 12 chains to an inch, it will represent an arc of 16×12 = 192 chains 2 miles and 3 furlongs radius; and similarly for other scales.

(3.) The limit of the radii of railway curves.

As a general rule the curves of a railway should be kept of as large a radius as possible consistently with the economical layingout of the line. One mile radius or upwards is quite unobjectionbut half-a-mile, or even less, will be sanctioned by Parliament

able;

in difficult cases, particularly near large stations, where the speed is usually slackened. In all sharp curves the outer rail requires to be raised above the level of the inner one, to counteract the effect of the centrifugal force. (See Formulæ in Section VII.)

PROBLEM I.

CASE I.

To determine the radius of a railway curve mechanically, that is, by the curve-rulers; the positions of the straight or tangential portions of the railway being given.

This problem admits of an indefinite number of solutions, but the curve adopted is generally that which falls on ground presenting the fewest natural or artificial obstructions, provided its radius be not too great. (See Art. 3.)

Let AB, CD be two straight portions of a railway, the positions of which are given by an accurate plan; and which, when prolonged, meet at T. Apply several of the series of curve-rulers (see Art. 2) to touch the line AB, without cutting it, at the points B, B', B", &c., and also to touch CD, in like manner, at the points C, c', c", &c. ; and let BC, BC', BC". &c., be the curves thus obtained. Then whichsoever of these

determined by multiplying the number on the curve-ruler by the number of chains per inch on the scale of the plan.

Example. If the curve BC be drawn by the curve-ruler, numbered 22, and be adopted as part of the line, and the scale of the plan be 5 chains to an inch, required the radius of the curve.

22 × 5 110 chains =

=

1 mile 3 fur., the radius required.

NOTE. By this method the radii of railway curves are commonly found in practice; but since a circular arc of large radius apparently coincides with its