When sharp curves are unavoidable, they should, if possible, be located near stopping-places. They should not be placed on a steep slope, on account of the double resistance which would then be caused to trains ascending, and the increased danger of running off to trains rapidly descending. But if such location on a long slope be unavoidable, the grade should be flattened along the curve, and the difference applied to the straight portions. Curves should not be in deep cutting, where the impossibility of seeing far ahead might cause collisions, but on the parts in embankment, or on the surface.

The increased velocities of the more recent railroads have greatly lessened the permissible smallness of the radii of curves. For the usual speeds employed on the English railways, it is recommended, that the minimum radius should be one mile. On the Baltimore and Ohio railroad, however, one of the earliest in the United States, there are several curves of 400 feet radius, (141°) and one of 318 feet, (18°) over which locomotives pass without difficulty at a speed of 15 miles per hour.

The minimum in France, allowed by "L'Administration des Ponts et Chaussées," is 2700 feet; or about 2°. The minimum curve upon the Hudson River railroad has a radius of 2062 feet=23°.

By the Parliamentary "Standing Orders" of 1846, a Railroad Company cannot diminish the radius of any curve to less than half a mile (2640 feet) without the special permission of Parliament.

2. WHAT RAILROADS OUGHT TO BE AS TO THEIR GRADES.

The question of the steepest grade admissible on a railroad is not one of practicability, as is often supposed, but only one of comparative economy. Locomotive engines can be made to ascend grades of almost unlimited steepness, by a proportionate increase of their power and adhesion, but their ascent becomes less and less useful in proportion as the grades become more and more steep. On an ascent of 19 feet to the mile, an engine can draw only about one-half its load on a level; at 38 feet to the mile, only one-third, and so on, (adopting the usual, though insufficient, ratio of 8 lbs. to the ton, or 1 to 280, as the resistance on a level) since, on this supposition, if the railroad rises 1 foot in 280, an additional force of 8 lbs. will be required to draw one ton up this ascent, (see page 32) and therefore double the former force will be needed to draw the fornier load. Only half the load, therefore, could be drawn by the same force; or that amount of power which could draw a load a mile on a level, would be exhausted in drawing it half a mile up this ascent.*

*The precise ratio between the total resistance on a level road, and that on any ascent, and therefore between the comparative loads which can be carried on each, may be found by the proportion which will now be investigated.

The loads on a level, and on an ascent, are in the inverse ratio of the resistance thereon: i. e.

The load on the level is to the load up the ascent, as the total resistance on the ascent is to the resistance on the level.

The resistance on the ascent is compounded of that of friction, &c. on the level, and that of gravity, which is such a part of the whole load, as the height of the ascent is of its length, as shown on page 32.

Adopting the more correct ratio of 101 lbs. per ton, or 1 to 218, as the resistance at the usual freight speed of 12 miles per hour, (see page 266) it would require an ascent of 24 feet per mile to double it, 48 feet to triple it, and so on. When the resistance is increased to 20 lbs. per ton, or 1 to 112, (as in the case at high velocities) an ascent of 47 feet per mile is required to double it; and a resistance of 30 lbs. per ton corresponds to an ascent of 70 feet.

These results show that heavy grades are proportionally less injurious on a road where great speed is employed, with correspondingly great resistances, though the absolute loss of power caused by them remains the same. The late discovery, that the resistances at even slow rates of travel are greater than had been supposed, lessens greatly the objections to heavy grades, and shows them to be relatively much less injurious than had been imag... ined, seeing that so much greater an ascent is required to double the resistance. Besides, a small diminution in the

Let then f= Resistance (in lbs. per ton) on a level.

h = Ascent in feet per mile; and

h

5280

= Inclination.

When the motive power is a Locomotive Engine, as is usual, its weight must be included in the "Load on level," used in the calculation, and

finally subtracted from the resulting "Load up ascent."

Example.-Let the weight of the cars drawn on a level, at 12 miles per hour, be 447 tous; the engine 2), and the tender 14 tous: required

velocity of the train would compensate for the increased resistance of quite a steep grade.

This

The cost of draught on a railroad is nearly as the power employed, so that it will cost nearly twice as much to carry a load on a railroad with an ascending grade of 24 feet to the mile, as to carry it on a level route. consideration will therefore justify large expenditures upon the excavations, embankments, &c., of a railroad, with a view of reducing its grades. The propriety of such expenditures is to be determined by comparing the annual interest of the amount with the annual saving of power ever after, in drawing the expected loads over the flattened road.

But, on the other hand, this principle may be carried to excess. These great expenses for graduation should be incurred only when maximum loads are to be constantly carried at high speeds, as on important leading lines of great traffic. Much steeper grades, than would be otherwise allowable, may be adopted on roads on which maximum loads are not often carried, and on which the trains are required for public convenience to go often, and will therefore generally go light. The engine may be able to draw 400 tons on a level, and may seldom have more than 100 to draw. In such cases the true economy is,

the load which the same power can draw up an ascent of 10 feet per mile.

For the method of calculating the tractive power of locomotives, set

page 325.

not to go to great expense in order to reduce the grades. below such a degree of steepness as would permit the engines to draw up their usual small loads; nor to attempt to make a very level road, on which the engines could do a great deal, but would have very little to do. The same reasoning applies to railroads between places furnishing but a moderate amount of travel, such as the thinly settled parts of this country. Should the travel subsequently greatly increase, in an unanticipated degree, more frequent light trains could be sent. The enormous expenditures sometimes made in such situations to make a perfect road, have been too great for the scanty travel to pay interest upon, and have discouraged the proper construction of such as would have been really profitable.

A great reduction of the first cost of a railroad may often be made, without much increasing its subsequent expenses; inasmuch as the capital expended in the graduation of a road has averaged, in England, fifteen times the cost of the locomotive power; and as the daily cost of transit, due to this last, is also very small. Locomotive power forms only about one-third of the whole working expenses of a road; and only a part of this, say one-half, is likely to be affected by the grades; so that there is only onesixth of the whole working expenses, which can be saved by making a road theoretically perfect in grades; a small consideration for the interest of the extra capital, unless the traffic is likely to be continued, regular, and very heavy.

In brief, first determine precisely what is wanted. If the best possible road would be justified by the importance of the traffic, make it as perfect (i. e. as straight, level, and unyielding) as possible, so that it can accomplish the greatest amount of labor in the least time and