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836 out of London with a ruling gradient of 1 in 1320. Other engineers, however, such as Joseph Locke, cheapened the cost of construction by admitting long slopes of 1 in 80 or 70. One of the steepest gradients in England on an important line is the Lickey incline at Bromsgrove, on the Midland railway between Birmingham and Gloucester, where the slope is 1 in 37 for two miles. The maximum gradient possible depends on climatic conditions, a dry climate being the most favourable. The theoretical limit is about 1 in 16; between 1 in 20 and 1 in 16 a steam locomotive depending on the adhesion between its wheels and the rails can only haul about its own weight. In practice the gradient should not exceed 1 in 22½, and even that is too steep, since theoretical conditions cannot always be realized; a wet rail will reduce the adhesion, and the gradients must be such that some paying load can be hauled in all weathers. When an engineer has to construct a railway up a hill having a still steeper slope, he must secure practicable gradients by laying out the line in ascending spirals, if necessary tunnelling into the hill, as on the St Gothard railway, or in a series of zigzags, or he must resort to a rack or a cable railway.

Rack Railways.—In rack railways a cog-wheel on the engine engages in a toothed rack which forms part of the permanent way. The earliest arrangement of this kind was patented by John Blenkinsop, of the Middleton Colliery, near Leeds, in 1811, an an engine built on his plan by Mathew Murray, also of Leeds, began in 1812 to haul coals from Middleton to Leeds over a line 3½ m. long. Blenkinsop placed the teeth on the outer side of one of the running rails, and his reason for adopting a rack was the belief that an engine with smooth wheels running on smooth rails would not have sufficient adhesion to draw the load required. It was not till more than half a century later that an American, Sylvester Marsh, employed the rack system for the purpose of enabling trains to surmount steep slopes on the Mount Washinton railway, where the maximum gradient was nearly 1 in 2½. In this case the rack had pin teeth carried in a pair of angle bars. The subsequent development of rack railways is especially associated with a Swiss engineer, Nicholas Riggenbach, and his pupil Roman Abt, and the forms of rack introduced by them are those most commonly used. That of the latter is multiple, several rack-plates being placed parallel to each other, and the teeth break joint at ½, ⅓ or ¼ of their pitch, according to the number of rack-plates. In this way smoothness of working is ensured, the cog-wheel being constantly in action with the rack. Abt also developed the plan of combining rack and adhesional working; the engine working by adhesion alone on the gentler slopes but by both adhesion and the rack on the steeper ones. On such lines the beginning of a rack section is provided with a piece of rack mounted on springs, so that the pinions of the engine engage smoothly with the teeth. Racks of this type usually become impracticable for gradients steeper than 1 in 4, partly because of the excessive weight of the engine required and partly because of the tendency of the cog-wheel to mount the rack. The Locher rack, employed on the Mount Pilatus railway, where the steepest gradient is nearly 1 in 2, is double, with vertical teeth on each side, while in the Strub rack, used on the Jungfrau line, the teeth are cut in the head of a rail of the ordinary Vignoles type.

Cable Railways.—For surmounting still steeper slopes, cable railways may be employed. Of these there are two main systems: (1) a continuous cable is carried over two main drums at each end of the line, and the motion is derived either (a) from the weight of the descending load or (b) from a motor acting on one of the main drums; (2) each end of the cable is attached to wagons, one set of which accordingly ascends as the other descends. The weight required to cause the downward motion is obtained either by means of the material which has to be transported to the bottom of the hill or by water ballast, while to aid and regulate the motion generally steam or electric motors are arranged to act on the main drums, round which the cable is passed with a sufficient number of turns to prevent slipping. When water ballast is employed the water is filled into a tank in the bottom of the wagon or car, its quantity, if passengers are carried, being regulated by the number ascending or descending.

Curves.—The curves on railways are either simple, when they consist of a portion of the circumference of a single circle, or compound, when they are made up of portions of the circumference of two or more circles of different radius. Reverse curves are compound curves in which the components are of contrary flexure, like the letter S; strictly the term is only applicable when the two portions follow directly one on the other, but it is sometimes used of cases in which they are separated by a “tangent” or portion of straight line. In Great Britain the curvature is defined by stating the length of the radius, expressed in chains (1 chain&thinsp;=&thinsp;66 ft.), in America by stating the angle subtended by a chord 100 ft. long; the measurements in both methods are referred to the central line of the track. The radius of a 1-degree curve is 5730 ft., or about 86⅔ chains, of a 15-degree curve 383 ft. or rather less than 6 chains; the former is reckoned easy, the latter very sharp, at least for main lines on the standard gauge. On some of the earlier, English main lines no curves were constructed of a less radius than a mile (80 chains), except at places where the speed was likely to be low, but in later practice the radius is sometimes reduced to 40 or 30 chains, even on high-speed passenger lines.

When a train is running round a curve the centrifugal force which comes into play tends to make its wheel-flanges press against the outer rail, or even to capsize it. If this pressure is not relieved in some way, the train may be derailed either (1) by “climbing” the outer rail, with injury to that rail and, generally, to the corresponding, wheel-flanges; (2) by overturning about the outer rail as a hinge, possibly without injury to rails or wheels; or (3) by forcing the outer rail outwards, occasionally to the extent of shearing the spikes that hold it down at the curve, thus spreading or destroying the track. In any case the details depend upon whether the vehicle concerned is an engine, a wagon or a passenger coach, and upon whether it runs on bogie-trucks or not. If it is an engine, particular attention must be directed to the type, weight, arrangement of wheels and height of centre of gravity above rail level. In considering the forces that produce derailment the total mass of the vehicle or locomotive may be supposed to be concentrated at its centre of gravity. Two lines may be drawn from this point, one to each of the two rails, in a plane normal to the rails, and the ends of these lines, where they meet the rails, may be joined to complete a triangle, which may conveniently be regarded as a rigid frame resting on the rails. As the vehicle sweeps round the curve the centre of gravity tends to be thrown outwards, like a stone from a horizontal sling. The vertical pressure of the frame upon the outer rail is thus increased, while its vertical pressure on the inner rail is diminished. Simultaneously the frame as a whole tends to slide horizontally athwart the rails, from the inner towards the outer rail, urged by the same centri fugal forces. This sliding movement is resisted by placing a check rail on the inner side of the inner rail, to take the lateral thrust of the wheels on that side. It is also resisted in part by the conicity of the wheels, which converts the lateral force partly into a vertical force, thus enabling gravity to exert a restoring influence. When the lateral forces are too great to be controlled “climbing” occurs. Accidents due to, simple climbing are, however, exceedingly rare, and are usually found associated with a faulty track, with “plunging” movements of the locomotive or vehicle, or with a “tight gauge” at curves or points.

From consideration of the rigid triangular frame described above, it is clear that the “overturning” force acts horizontally from the centre of gravity, and that the length of its lever arm is, at any instant, the vertical distance from the centre of gravity to the level of the outer rail. This is true whatever be the tilt of the vehicle at that instant. The restoring force exerted by gravity acts in a vertical line from the centre of gravity; and the length of its lever arm is the horizontal distance between this vertical line and the outer rail. If therefore the, outer rail is laid at a level above that of the inner rail at the curve, overturning will be resisted more than would be the case if both rails were in the same horizontal plane, since the tilting of the vehicle due to this “superelevation” diminishes the overturning moment, and also increases the restoring moment, by shortening in the one case and lengthening in the other the lever arms at which the respective forces act. The amount of super elevation required to prevent derailment at a curve can be calculated under perfect running conditions, given the radius of curvature, the weight of the vehicle, the height of the centre of gravity, the distance between the rails, and the speed; but great experience