enough to bring the flange of the wheel in contact with the rail.

Now we have seen above, that the necessary lateral deviation is expressed by

و سلام

If, therefore, the waggons have, for instance, a play of 2 in. on the way altogether; that is to say, if, in their regular position, the flanges of the wheels keep on each side at a distance of 1 in. from the rail, the greatest value of the deviation u, must always be less than 1 in. By that greatest value of , we mean the deviation on the most abrupt curve of the line. Consequently, putting for r the radius of that curve, and for its maximum, 1 in. or μ of a foot, the equation will give the greatest value that can be given to the quantity a, or the least

For instance, on a line, the most abrupt curve of which has 500 ft. radius, with waggons having wheels of 3 ft. diameter, and a play of 1 in. on each side of the way, the equation shows that the least inclination one ought to give to the tire of wheel is; but a more considerable inclination will answer, a fortiori.

On the Liverpool and Manchester Railway, the most abrupt curve, which is the one at the entrance of Manchester, has a radius of 858 ft. This results

a conical inclination of, and this would answer in all cases; but having said that a greater inclination will fulfil the same object, we are free to adopt a greater inclination, if it suits other purposes better.

It is customary to give an inclination of 4. The motive for making it so considerable, is to prevent all possibility of the flange rubbing against the rail, either in case of a strong side-wind, or in case of some fortuitous defect in the level of the rails, by which the waggons would be thrown on the lower rail. Having seen above that, with an inclination of, there would be no danger of the flange rubbing in the curves, that danger will be still more impossible with an inclination of 4.

We conclude that, with wheels having that inclination, the surplus of elevation of the rail which we have determined above, will correct the first species of resistance of the curves without creating the second, and that, consequently, the train will pass over the curves without any diminution of speed.

§ 2. A Practical Table of the Surplus of Elevation of the outward Rail in Curves, in order to annul the effect of those Curves.

From what has been said, the surplus of elevation that must be given to the outward rail in the curves, is determined by the following formulæ :

U

In this equation the signs have the following value :

D, diameter of the wheel, expressed in feet.

r, radius of the curve, expressed in the same

manner.

e, half of the width of the way, expressed the same. V, average velocity that is to be given to the motion, expressed in feet per second.

g, accelerating force of gravitation, expressed in feet per second, or g = 32 ft.

y, surplus of elevation to be given to the outward rail of the curve over the inward rail, expressed in feet and decimals of feet.

Solving these formulæ in the most usual cases on railways, we make out the following table, which dispenses with all calculations in that respect.

A PRACTICAL TABLE OF THE SURPLUS OF ELEVATION TO BE GIVEN TO THE OUTWARD RAIL IN THE CURVES, IN ORDER TO ANNUL THE RETARDING EFFECT OF THE CURVES.

ARTICLE II.

OF THE INCLINED PLANES.

§ 1. Of the Resistance of the Trains on Inclined Planes.

Inclined planes are a great obstacle to the motion on railways.

As soon as the trains reach these inclined planes, they offer a considerable surplus of resistance, on account of the gravity of the total mass that must be drawn up the plane.

On

Let us suppose a train of 100 t. drawn by an engine. Having seen that on a level the friction of the waggons produces a resistance of 8 lbs. per ton, the power required of the engine will be 800 lbs., when travelling on a level. But let us suppose the same train ascending an inclined plane at that plane, besides the resistance owing to the friction of the waggons, a fresh resistance occurs, which is the gravity of the total mass in motion on the plane. That gravity is the force by virtue of which the train would roll back if it were not retained; and it is equal to the weight of the mass divided by the number that indicates the inclination of the plane. If, therefore, in this case, the load of 100 t. is

100

drawn by an engine weighing 10 t., the total mass placed on the inclined plane will be 110 t. or 246,400 lbs.; and thus its gravity on the inclined plane, at, will be 246.400 lbs. = 2,464 lbs. The surplus of traction required of the engine, on account of that circumstance, is, therefore, 2,464 lbs., and, as we have seen that on a level 1 t. load is represented by 8 lbs. traction, we also see that those 2,464 lbs. represent the resistance that would be offered by a load of 308 t. on a level. Consequently, the engine, which, before, drew 100 t. must now draw 408 t., or at least must exert the same effort as if it drew 408 t. on a level.

This is the manner in which the calculation of the resistance on inclined planes must be established; and we have entered into those particulars, because it frequently happens that, in making the calculation, the gravity of the load is alone considered, without taking into account the gravity of the engine, which ought also to enter for its share.

In speaking of the fuel, we shall see that the inclined planes of the Liverpool Railway, which at first sight appear quite insignificant, oblige, however, the engines to a surplus of work, which amounts to a sixth part of what they would have to do on a level. By this we see how important it is, in establishing a railway, to keep it on as perfect a level as possible. It frequently happens that, by avoiding to level a part of the road, that is to say, to cut through a hill, or to form an embank