=255. Train Resistances.=—It is a fact confirmed by constant daily experience that, however nicely the machine impelling the railroad train or the tracks supporting the cars may be built, considerable frictional and other resistance is offered to the movement of the train when the latter passes over a perfectly level and straight track. A considerable portion of the cost of transportation is expended in overcoming this resistance. When the line fails to be either level or straight other resistances of magnitude are developed; they are called the resistances of grades and curves: and it is the business of the civil engineer so to design the railroad as to reduce these two classes of resistance to an absolute minimum, in view of certain other conditions which must be concurrently maintained. =256. Grades.=—The grade of a railroad is expressed usually in this country by the number of feet through which 100 feet of length of line rises or falls, or by some expression equivalent to that. If, for instance, the line rises 1.5 or 2 feet in 100, it is said to have an ascending grade of 1.5 or 2 per cent. Or if the line falls the same amount in the same length, it is said to have a descending grade of 1.5 or 2 per cent. It is evident that a grade which descends in one direction would be an ascending grade for trains moving in the opposite direction, so that grades favoring traffic in one direction oppose it in the other. Hence, other things being equal, that road is the most advantageous for the movement of trains which has the least grade. The grades of railroads seldom exceed 2 or 2.5 per cent, although, as will presently be shown, there are some striking exceptions to that general observation. The actual angles of inclination of railroad tracks from a horizontal line are therefore as angles very small, but their disadvantages for traffic increase rapidly. A simple principle in mechanics shows that if the railroad train with a weight _W_ moves up a 2 per cent grade, one component of the train weight acts directly against the tractive force of the locomotive or other motive power. If _a_ is the angle of inclination of the track to a horizontal line, this opposing component will have the value _W_ sin _a_. When angles are small their sines are essentially equal to their tangents. Hence, in this case, sin _a_ would have the value .02 or ¹/₅₀ of the train weight. If the weight of the train were 500 tons, which is a rather light train for the present time, this opposing force would be 10 tons, or 20,000 pounds, which, as we shall see later on, is more than one half of the total tractive force of any but the heaviest locomotives built at the present day. This simple instance shows the advantage of keeping railroad grades down to the lowest practicable values.