train is small compared with the Total Energy to be expended on them.

4. The nearer the uniform grade approaches the “Maximum grade," the more injurious do any breaks become; and the only point in connection with the "Maximum grade," where an increase in the rate is allowable, is the insertion of a 'Momentum grade" at its lower end.

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5. Breaks in a grade are more injurious to slow than to fast traffic-as may be seen from the Table of Equivalent Heights -e.g., an increase in elevation of 20 feet reduces the velocity from 30 to 12 miles per hour, while a velocity of 60 m. p. h. is only reduced to about 55 miles per hour.

6. Be careful in inserting Momentum grades that they will not be such as to cause the velocity at any point to exceed the safe limit. A difference in elevation of about 30 feet between the Broken and the Uniform grade should generally be taken as a limit.

29. Another point to be considered, which we have not yet referred to, is the increase in Liability to Danger of Breaking-train and Derailment to which an undulating grade gives rise. For, suppose in Fig. 2 we have a train running up the grade from A to B: as soon as the engine is over the summit the pull on the draw-bar becomes enormously increased, and similarly with the car-couplings throughout the entire train; so that, unless the greatest care is taken in applying the brakes, the train runs a very great risk of being broken in two. Similarly, in such a hollow as E, the cars near the centre of the train are liable to get terribly jammed together, thereby greatly increasing the chances of Derail

ment.

Vertical curves reduce these dangers considerably, but not entirely.

It must be remembered that it is not in the least necessary that one of the grades should be an up-grade and the other a down-grade: it is the difference in the rate of grade that has to be looked out for. (See Sec. 100.)

30. In Fig. 3, let ACB and ADB represent two different routes between A and B, the total Rise and Fall between the two points in each case being the same. The amount of work done in hauling the train from A to B by way of C will, supposing we are dealing with grades so long that the ques

tion of "Momentum Grades" may be ignored, be then practically the same as by way of D. Similarly, if such a point as H in Fig. 4 has to be reached, the work done in hauling the train along the uniform grade EH will be practically the same as by way of FG. It is not the amount of work done on the grades themselves that has to be considered, but the amount of extra work which is uselessly done by a heavy engine hauling a large surplus of dead-weight (due to its own size) over

grades where a lighter engine could have hauled the train equally well. If each of the divisions EF, FG, and GH were a suitable length for one engine to work, the lower route would then be as economical probably as regards Operating Expenses as the higher. Besides this, we have the increased

H

E

FIG. 4.

consumption of fuel, before referred to, which always accompanies variations in grade.

If we make each of the divisions along the lower route from E to H of such a length as to keep the engine employed on each fairly busy,-using a different engine on each division,the lower route is then as economical as can be wished for, but otherwise the upper route has the advantage.

31. Now the average length of an Engine-stage may be considered to be about 100 miles, which is of course too long to enable us to work the lower route in the manner described above. We may often, however, by adopting a Pusher-grade, even at a point where at first it appears unnecessary, make a decided improvement in the economy of our grades. The length of this grade, if the Pusher is to be kept steadily employed, depends of course on the number of trains to be taken up it each day: if there are four trains a day the engine will be kept sufficiently at work if the length of the grade is only 12 miles. As to the rate of grade which may be adopted in such cases as this, Mr. Wellington gives the following Table, which is suitable for average Consolidation Engines, the coefficient of adhesion being taken at 0.25:

32. Maximum Curvature.-In countries where construction is comparatively easy, it is often the custom to select a certain degree of curvature which is not to be exceeded. The question of the speed required to be maintained is the main one which arises in this case. Wear and tear of rails and rollingstock is also an important factor. The question of resistance -at ordinary speeds-is comparatively unimportant, since at a speed of 25 miles per hour a 10° curve only offers the resistance of about a 0.3 p. c. grade. In rough country it is impossible to fix a "maximum," "for the additional cost of construction which the adoption of a limiting-grade might involve would perhaps be an inconceivably greater consideration than the loss of a few seconds-or possibly minutes-in time. As regards the question of the Safe Speed on Curves, it is diffi

cult to lay down any law, but it is supposed to vary inversely as the square root of the radius. Thus if we assume that 40 miles per hour is a safe speed on a 2° curve, the speed should be limited to 20 m. p. h. on an 8° curve and to 14 m. p. h. on a 16° curve. The chances of derailment and the wear and tear of rolling-stock and rails are decreased materially by the use of Transition curves. (See Sec. 96)

33. It is almost unnecessary to refer to the subject of Reverse Curves. In Station-yards, where the speeds are insignificant, their use is sometimes advisable; but on the Main Track an intervening tangent of at least 200 feet in length should be regarded as an absolute necessity. A fault much more frequently found is the insertion of a short tangent between two curves of the same direction. Getting on to a tangent from a curve is as hard work as getting on to a curve from a tangent; and since it is at the P. C. and P. T. that the curve gives its maximum resistance, the curves should at least be compounded so as to make the radius of curvature at all points as uniform as possible, for in each case the total amount of curvature will be the same. Another point to be remembered-though it is not often that it can be applied-is, that a road which has its curves at points where the speed is comparatively low has a decided advantage over one in which the curves are located at places where a high speed is required to be maintained. Thus, if a certain amount of curvature has to be got in, in such a place as DEF in Fig. 2, it should be arranged if possible so that the curvature at D and F will be sharper than at E. Curvature should also be avoided as much as possible at all points where a stoppage is required, for on starting, the resistance due to the curvature is a great consideration, and, as we saw in Sec. 6 and Diagram I, will probably make it as difficult for the train to start as a decided up-grade.

34. We have now dealt in a more or less superficial way with most of the mechanical problems which arise in connection with railroad trains; but it is convenient, for the sake of more readily comparing the value of the various resistances to passenger and freight trains at average speeds, to tabulate their mean values (as given by Prof. Jameson) as follows:

TABLE SHOWING COMPARATIVE VALUES OF RESISTANCES AS REGARDS WORK DONE.

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Rise and Fall" of course means in one direction only, and is so stated in order to take account of the Rise when running in the opposite direction. Thus in Fig. 3 the total Rise and Fall between A and B by either route equals 710 feet.

COST OF OPERATING.

35. The expense involved in overcoming the resistances referred to in Sec. 34 is not proportional to the amount of work which is performed on account of them. For instance, it is found by experience that hauling a train over one mile of level track costs on an average about the same as 150 feet of rise and fall,—not of 25 feet, as given in the last table. Similarly, with curvature, the operating of one mile of level track is found to cost the same as about 900° of curvature (not 600°); so that as regards operating-expenses the table given in Sec. 34 becomes

As soon, then, as we know the expense of operating one mile of level track, we can by means of this table find the probable cost of working any certain grade or any given amount of curvature.

36. Taking $1.00—it is probably nearer 90 cts.—as the average cost of operating one mile of level track on American Railroads for each train that runs over it (and returns) each day, we can make this our unit of operating-expenses and