The late A. M. Wellington, in giving a detailed solution of a problem substantially like the above, declared that he had taken velocity and dynamometer records in hundreds of cases of trains that were operated substantially as above, and that he had found that for all practical purposes the drawbar pull was constant, whether the velocity was great or small, and that the velocity at the foot of the sag or at the summit of the hump was substantially in accordance with the theoretical figures obtained for the particular case. In a paper read before the American Society of Civil Engineers on December 3, 1902, by Mr. A. C. Dennis, the statement was made that, as a result of tests aggregating thousands of miles of train operation, he found that freight-train resistance, when duly corrected for existing curvature and grade and for change in velocity, was substantially a uniform quantity at about 4.7 pounds per ton for loaded trains between the velocities of 7 and 35 miles per hour. Since the velocities in the above example are well within these limits, there would be little or no error due to variation of tractive resistances. Assume that the train in the above problem weighs 1500 tons. Let us assume that it approached the point A on a level track, and that it was moving at a velocity of 15 miles per hour, which is at the rate of 22 feet per second. Assuming that the tractive resistance is 4.7 pounds per ton, we would have, as the total horse-power developed at the speed of 15 miles per hour, 1500x4.7x22 = 282 H.P.

According to the assumption of a uniform draw-bar pull, when the train reached the bottom of the sag it would be moving at a velocity of 28.2 miles per hour, and it must therefore be developing 530 H.P. When it has moved up the succeeding grade and has reached the summit of the hump, the velocity is assumed to be 10.5 miles per hour instead of 15, and the horse-power required at this velocity will be only 197.4. Although the horse-power developed by a locomotive may vary between rather wide limits, the range of this variation is subject to definite limitations. At a very low velocity the tractive power is absolutely limited by the frictional resistance between the driving-wheels and the rails. Although the coefficient of friction will not ordinarily exceed 25%, there are some cases where, by the use of sand, a coefficient approximating one-third may be obtained. Therefore the weight on the drivers multiplied by 25% is usually a limiting measure of the tractive power of the locomotive. At very low velocities the maximum horsepower of the locomotive is therefore limited by the product of the maximum tractive power and the velocity which the locomotive can develop. At some speed, which is usually over 10 to 15 miles per hour, it not only becomes impossible for the locomotive to develop steam fast enough to supply the cylinders at full stroke, but it also becomes far more effective to use the steam expansively. The maximum tractive force which can be developed by an engine for one complete revolution of a driver equals Theoretical tractive force (diam. piston)2 x av. steam-pr. x stroke (13) = diameter of drivers.

The effective steam-pressure is considerably less than this, and none of the above quantities are variable except the pressure. If the effective steam-pressure in the cylinder is reduced, as must be the case when the steam is used expansively, then the effective tractive power is unquestionably reduced. In spite of the reduction in effective steam-pressure, it is possible that the speed may become so high that the horse-power developed is greater, in spite of the reduced draw-bar power, than it was before. Nevertheless, the draw-bar pull certainly does decrease with increased velocity. The speed at which it will begin to decrease depends on the ability of the boiler to develop steam rapidly. In Fig. 26 is shown in a diagram the reduction in tractive power with increase of velocity of the consolidation locomotive with which Mr. A. C. Dennis made the tests above referred to. It will be noticed in this particular case that the draw-bar pull commenced to decrease immediately, and that at 14 miles per hour the tractive force had reduced to 75%.

## 127. Utilization Of Table XX

Locomotive engineers very soon learned to utilize the advantage of "a run at the hill." and found that whenever they were able to approach a hill with a high velocity they would be able to draw up that hill a considerably heavier train than could be hauled if they started from rest or at a low velocity at the bottom of the hill. This advantage, however, is limited by the length of the hill, and it really becomes a question of the difference of elevation which can be surmounted by virtue of the kinetic energy stored in the train when it reaches the bottom of the hill. As the train climbs the hill and its velocity diminishes, the tractive force will increase rather than diminish, the tractive resistance will diminish rather than increase (assuming that the velocity does not decrease to less than 7 miles per hour), and therefore the kinetic energy can all be utilized in overcoming the elevation. About the only exception to this occurs when a freight-train has been forced to attain such a high velocity at the bottom of a hill that the reserve boiler-power has been overtaxed, and the boiler-pressure falls because the boiler is unable to produce steam with sufficient rapidity, but this will be largely a matter of the way the engine is handled. In a very simple case, such as the mere insertion of a hump either on an otherwise level track or on an otherwise uniform grade, such as is illustrated in Fig. 25 (a) or (b), whenever we can rely on a train reaching that hump with a sufficient velocity, and with the engine doing an amount of work which would carry it along the uniform grade at that velocity, Table XX will show whether that hump can be surmounted so that the velocity at the summit of the hump will still be within practicable limits, say 10 miles per hour. The other case (c) in Fig. 25 cannot always be determined accurately by this table, although the table may be depended on to give an approximate result, unless the case is very extreme. If a sag is very deep, one of several things may happen. First, the velocity at or near the bottom of the sag may become so great that steam must be shut off and brakes applied in order to prevent the train from attaining an objectionably high velocity. In such cases the table is not supposed to apply, since an express condition of the table is that the tractive force exerted by the engine is uniform. Second, even if it is attempted to operate the engine so that the tractive force is uniform, it may become impossible, as explained above, for the boiler to make steam fast enough to develop such power. If that full amount of power is not developed at the bottom of the sag, then the full amount of kinetic energy will not be developed, which will be necessary to permit the train to surmount the steeper grade and reach the upper end of the sag with its original velocity. Whether this will be the case can best be determined by means of momentum diagrams, such as will be described later.

(a) Hump in track otherwise level.

(b) Hump on a grade otherwise uniform.

(c) Sag on a grade otherwise uniform.

Fig. 25. Sags and humps on grades otherwise uniform.