line, then the flanches of the wheels, are the only guides to keep the carriages on the rails.

The degree of cone, generally given to the tire of the carriage wheels, is, to make the diameter, next the flanch, one inch larger than the diameter next the outside of the tire; the breadth being three and a half inches. In practice, it is likewise usual to keep the wheels at such a distance from each other upon the axles, that when travelling upon a straight line, the flanches on each side are about one inch from the rail. Supposing the wheels three feet diameter, which is the general size, the curve which such wheels would describe, beforethe flanches would rub against the rails, is thus found.

1.05

3.5

The extreme lateral motion, generally given to the wheels upon the rails, is from one and a half to two inches, the difference of diameter on the whole breadth of three and a half inches is one inch; therefore, suppose the extent of lateral motion, before the flanch rubs against the rail, to be 1·05 inch, then 3, the difference of diameter in 1.05 inches; consequently, when the flanch is just rubbing against the side of the rail, one of the wheels is traversing a circle 36.3 inches in diameter, and the other, a circle equal to 36 inches. Now, the line which these wheels would describe, the width between the rails being fifty-six inches, would be a curve with a radius of 565 feet, viz., as 3 : 36 :: 56 : 565 feet; and as this is a curve of a less radius than any which ought to occur upon a railroad, we find that a cone of an inch, in the breadth of three and a half inches, given to carriage wheels of a diameter equal to three feet, is sufficient in all ordinary curves upon a railroad, to prevent the wheels rubbing against the inside of the rails.

We have now to consider, the effect of the centrifugal force of carriages, passing round curves of different radii.

As before stated, the tendency of carriages passing around the curves upon a railway, is to keep a tangential course; this is capable of being measured, and is a certain force, depending upon the velocity of motion, and the radius of the curve, the formula being well known.

Let w the weight of the carriages,

r the radius of the curve in feet,

v=the velocity of the carriages in feet per second, g=the accelerating force of gravity, or the velocity

which a body would acquire in falling a second of time.

Then the fractional part of the weight of the carriage,

representing the centrifugal force, will be G=w+

v2 gr

Thus, suppose the velocity of the carriage to be fifteen miles an hour, or twenty-two feet per second, and the radius of the curve 500 feet,

If, therefore, we elevate the outside rail of the railway, to such a height, above the inner rail, that we give to the axles of the carriages resting upon the two rails, such an inclination as will produce upon the carriages a gravitating force towards the centre of the curve, equal in amount to that of the centrifugal force outwards; there will neither be any tendency in the carriages to upset, or to press the wheels against the rails. Thus, upon the curve above named, if the outside rail be elevated to such a height above the inner rail, as will give to the axles of the carriages, such an inclination as, together with the difference of diameter of the wheels, where running upon the rails on the curve, will be equal to the one thirty-third part of the breadth between the rails, when the carriages travel at the rate of fifteen miles an hour; the gravitating force inwards will coun

teract that of the centrifugal force outwards, in passing round the curve.

Let w=the width of the railway, and s=the surplus of elevation given to the end of the axle of the wheel, on the outside of the curve, in inches, d-ď—the difference of diameter of the two wheels, where running

upon the rails in traversing the curve; then s

v2w

qr

dd the height, in inches, which the outside rail of the curve, should be elevated above that of the inner rail, to counteract the effect of the centrifugal force of the carriages passing round the curve.

We must now find the value of d-d', which will vary as the radius of the curves. To accomplish this, we have to obtain the difference of diameter of the two wheels, which, in the breadth, w, of the railway, will cause the wheels to describe a curve, the radius of which is r. Supposing the wheels thirty-six inches in diameter, D, this will be DwW

V2w

gr

; consequently, we have

D W

r

When, however, we once determine, and lay down the rails suitable for a given velocity, and the carriages move at a slower rate, the force of gravity will overbalance that of the centrifugal, and the effect will be, a tendency of the carriages to press the flanches of the wheels against the inside rails of the curve; and, on the contrary, if they move at a greater velocity, the tendency will be, to press the flanches against the outside rails. As the slower trains are generally more heavily laden, and when, consequently, any increase of friction will have a more powerful effect upon the moving power, than with lighter trains travelling at a quicker rate; it will be advisable, that the elevation of the out

side rail, should not be greater than to compensate for the centrifugal force, at the slower rate of travelling with heavy trains.

The following Table will shew the elevation to be given to the outside rail, of different radii, above that of the inner rail of such curve; so that the whole amount of centrifugal force, is balanced by that of the gravity of the load, towards the inside of the curve.

§ 16.-Plan of crossing Streets, and Turnpike Roads.

Having thus described the mode of executing a railway on ordinary ground, the plan of drainage, and the coating and laying the railway; we shall now describe the plan of crossing streets and roads. Fig. 10 is a section, and Fig. 11, Plate V., a plan, of crossing a street, or public highway, near a town; a a a, &c. is masonry, forming the foundation whereon the railway is to be

laid, and the drains, bb, for carrying off the water, and acting, likewise, as a receptacle for the dirt. The blocks cc, are placed upon the masonry, and upon the blocks, and attached to the chair, is a frame of cast iron, inclosing the rails, as shewn in Fig. 11, rr being the rails, e e, e e, the cast-iron frame, with the cross ribs, Ill, &c. Between the rails, rr, and the sides, e e, of the iron frame, there is just sufficient width for the flanch of the wheels of the carriages to pass along; and being open underneath to the drains bb, except where covered by the blocks cc, and chairs ff, Fig. 11, all the dirt falls down into the drains, and keeps the rails entirely free for the passage of the carriages.

The stones gg, gg, are penned close against the outer sides of the iron frame, and on a level with the top of it; so that, when carts, or other carriages, cross the rails, they are not subjected to any jolt. Between the stones gg, the road is paved in the ordinary way, and which, resting on the solid masonry, is kept very firm and level.

Fig. 12 shews the manner of passing common roads, where the passage is not very great. In this case, the masonry is not required, the blocks c c are set, as in the common way, upon the coating a a; a cast-iron frame, similar to that before described, and shewn in Fig. 11, is laid upon the blocks, for the purpose of protecting the rail, from the wheels of the carriages crossing the road; but in this case, there is no drain or receptacle for the water and dirt, as in the former cases, any dirt falling into the cavity, against the rails, being taken out by the attendants, when necessary. The road may be either paved on the surface, against the frame-work, or covered with broken stones, or with the common material of the road; in the latter case, however, the