Assume that the grade of a railroad in crossing a river valley includes a sag 5000 feet long and with a depth in the center of 40 feet. Assume that freight-trains would ordinarily approach this sag at a velocity of 20 miles per hour. The velocity head at 20 miles per hour, as found in Table XX, is 14.05 feet. Adding the depth of the sag, 40 feet, we would have the velocity head at the bottom of the sag, 54.05 feet, which corresponds to a velocity of over 39 miles per hour. The extensive adoption of automatic couplers and train-brakes have permitted the use of much higher freight-train speeds than were permissible some years ago. Even though it might be considered safe to run the train through the sag at a speed of nearly 40 miles per hour, it is unquestionable that a freight-engine could not develop steam fast enough to exert a constant draw-bar pull up to a speed of 39 miles per hour. There would therefore be a very considerable loss from the theoretical operation of such a sag as described above, and we must consider that the sag will not belong to class {A), at least for freight-trains. If a passenger-train approached this sag at a velocity of 30 miles per hour, the velocity head then being 31.60 feet, its velocity head at the bottom of the sag would be 71.60, which would correspond to a velocity of over 45 miles per hour. If the passenger-engine were so lightly loaded that its draw-bar pull at the top of the sag was quite small, and its boiler capacity was so large that it could develop this light draw-bar pull even at a speed of 45 miles per hour, then for such trains we could consider the sag as belonging to class (A), the harmless class. Assume that after an analysis of the character of the trains using the sag, we find there are six trains per day each way in the operation of which the sag should be classed as class (C), and eight other trains per day each way for which it should be classed as class (B). It should be noted that it is not essential to fill up the sag altogether and make it level. If we fill up only the lower 20 feet, which will not ordinarily cost more than one-fourth to one-third as much as filling up the upper 20 feet, the sag would probably become harmless for all classes of trains. We will therefore compute the value of reducing the depth of the sag 20 feet. We will have the added cost of operating this 20 feet as follows:

 Eight trains, class (B) 8 x20 x \$2.41 = \$385.60 Six " " (C) 6 x 20 x \$3.21 = \$385.20 Total annual saving......... \$770.80

Capitalizing this at 5%, we have \$15,416 as the justifiable expenditure to fill up the lower 20 feet of this sag. Of course the amount of earthwork required to make this fill can be readily computed. In the case of a new road we would have merely this additional cost to the original plan of construction. If such a plan is considered with the intention of improving an old line, the cost of raising the track, and all the added expense involved in maintaining the track so that traffic may continue to run over it, will have to be added as part of the cost of improvement. The cost of such an improvement is a comparatively simple matter to determine. The above demonstration, even though it is based on data which is approximate, is at least a measure of the value of the improvement, which is far better than having no measure at all. In applying the method outlined above to any particular case, the problem must be studied from the beginning with reference to all the available figures of cost as applied to the given road. The figures given above for the value of one foot of rise and fall of either class should not be used in general for all cases, and in fact should never be used except as an approximate method for computing the value of a change in the proposed location of a new road where there is no data on which to base more accurate calculations.