The amount of this may be computed with mathematical exactness. Since the tractive resistances are computed separately, we merely have to compute the tendency of the wheels to roll down the grade, or the resistances to pulling them up the grade, which are exactly equal when the frictional resistance is zero or when it is otherwise provided for. Assume that a ball or cylinder (Fig. 23) is being drawn up an inclined plane. If we represent its weight by W, as measured graphically by the line W in the figure, then N will measure the normal pressure against the plane, and G will measure the force required to draw it up the plane with a uniform velocity. It also measures the tendency of the weight to roll down the plane. From similar triangles we may write the proportion G:W::h:d or G =wh/d .....(1)

In the diagram d is very much larger than c, but, as will be shown, c is so nearly equal to d on all practicable railroad grades that there is no appreciable error in substituting c for d, and write the equation G = wh/c. But h/c equals the rate of grade. Therefore we have the very simple and mathematically exact relation that the grade resistance G = W times the rate of grade. In order to appreciate exactly the extent of the approximation in assuming that the slope distance equals its horizontal projection, the percentage of the slope distance to the horizontal projection is given in the tabular form on page 186. Incidentally the tabular form shows the amount of error involved when we measure with the tape lying on the ground instead of holding it horizontally. Since almost all railroad grades are less than 2% (where the error is but .02 of 1%) and anything in excess of 4% is unheard-of for normal construction, the error in the approximation is generally too small for practical consideration.

 Grade in per cent. 1 2 3 4 5 Slope dist. / hor. dist x 100... 100.005 100.02 100.045 100.08C 100.125 Grade in per cent. 6 7 8 9 10 Slope dist. /hor. dist. x 100........... 100.18C 100.245 100.319 100.404 100.499