In the simplest case a rise and fall of so many feet means a rise from the starting-point to a summit and a return to the same level, as is shown in Fig. 34. For instance, in Fig. 34 (a) is indicated a rise and fall of BD above the level AC, but (b) and (c) may be considered just as much cases of rise and fall. In (c) the line AB is actually an up grade, and yet we may consider it as a virtual drop. If a freight-train is moving up the grade in (b) and the engine is doing the work which will carry it steadily up the grade ADC, it encounters additional resistance on the extra grade AB, and must either work much harder or it will continue to lose velocity. If it has sufficient momentum to carry it over the point B it will then continue on the grade BC, which, although an up grade, is so much less than the grade DC that the engine will do more work than is required on such a grade and hence will gain velocity. This is essentially the same case as though a train were moving uniformly along the level (case a) and encountered the hump ABC. The case is essentially the same in (c).

Although AB is actually an up grade, it is so much less than the grade ADC that if a train were running up that grade with the engine doing an amount of work which would carry it uniformly up the grade ADC, the resistance on the lesser grade AB will be so much less that the train will actually gain velocity and acquire a momentum which will enable it to climb the still steeper grade BC, so that by the time it reaches C it will have practically the same velocity which it had at A. We are therefore justified in considering that whether the train passes over a hump which is superimposed on an otherwise uniform grade, whether level or not, or goes through a sag which occurs in what would otherwise be a uniform grade, we may consider that in all cases the train is encountering what we denominate as "rise and fall." When the line runs through a stretch of several miles with very light grades, all of which are well within the ruling grade, there is in general no possibility of doing anything which will favorably affect the grade. Practically all that we can do is to remove what is virtually a hump or a sag in what is otherwise a nearly uniform grade or level.

Types of rise and fall.

Fig. 34. Types of "rise and fall".