It is exceedingly difficult to obtain experimental data showing the resistance in pounds per ton which is due to curvature. Mr. J. F. Aspinall, an English engineer, who has made many elaborate experiments on train resistance, has commented on this difficulty substantially as follows: When the experimental train enters the curve the engine encounters the additional resistance first, which decreases its velocity slightly, and the draw-bar pull actually diminishes instead of increases. As the train gradually moves on to the curve, the draw-bar pull increases until it will settle to some definite value after the entire train is on the curve; but, unless the curve is very long or the speed very slow, the engine will begin to leave the curve very soon afterward. On the track on which Mr. Aspinall conducted his experiments, these conditions existed to such an extent that no reliable computations of the curve resistance were possible.

Mr. G. R. Henderson uses the value 0.5 pound per ton per degree of curve. This is based on the assumption that the resistance varies directly as the degree of curvature. Although precise figures for the curve resistance are so scarce, it is definitely known that the total curve resistance does not increase as fast as the degree of curve. While the values given by Mr. Henderson's formula may be sufficiently precise for ordinary easy curvature, the application of such a figure to the curves of 90' radius on the New York Elevated road would mean a resistance due to curvature alone of about 34 pounds per ton. The curve resistance on these curves is far less than this figure. Although this is a very extreme case, it is a valuable check, since it shows the tendency of the increase of the resistance with an increase in the degree of the curve. This conclusion is also corroborated by the theoretical considerations already given, that the portion of the curvature resistance which is due to longitudinal slipping is absolutely independent of the radius. It is quite probable that the curvature resistance on sharp curves is also dependent on the velocity of the train, but, unfortunately, there is no experimental data by which such a conclusion can be definitely corroborated.

Searles makes an allowance of 0.448 pound per degree of curve per gross ton of 2240 pounds. He does not state the derivation of the value, nor how a value is obtained to the third significant figure, but, considering that such a value is the equivalent of precisely 0.4 pound per ton of 2000 pounds or a frictional coefficient of precisely .0002, it is possible that the apparently precise value may be based on a comparatively loose approximation.

## 119. Brake Resistances

The fact that grades may be so steep that they cannot be safely operated, when moving down the grade, without the use of brakes has been referred to in § 117. The energy consumed by brakes is hopelessly lost without any compensation. The kinetic energy possessed by the train is transformed into heat. All such energy is wasted and, in addition to this, a very considerable amount of steam is drawn from the boiler to operate the air-brakes which consume the power already developed. When trains are required to make frequent stops and yet maintain a high average speed, a considerable amount of power is consumed in applying the brakes. It has been demonstrated that engines, drawing trains in suburban service, making frequent stops and yet developing high speed between stops, will consume a very large proportion of the total power developed by the use of brakes. The brakes consume the power already developed and stored in the train as kinetic or potential energy, while the operation of the brakes requires additional steam-power from the engine. It should therefore not be forgotten that, in some kinds of service especially, the power required from the locomotive may be many times the amount of power which is required merely to overcome mere track and grade resistance.