It does not need proof that the sharper the curvature the greater will be the tractive force required. The rail wear and also the general wear and tear on road-bed and track per foot of length will also be increased. If we attempted to establish a relation between operating expenses and the radius of curvature, we would also have to consider the total length of the curve before we could determine the true effect on the operating expenses of any particular curve. A method of calculation, which is much more simple and which is sufficiently accurate for the purpose, may be made by establishing a relation between operating expenses and the number of degrees of central angle in a curve. The outline of the method is as follows:

(1) It has been found that if two tangents, which make an angle J, are connected by a curve of large radius, such as the curve AB, the total cost in operating expenses for the curve AB will not be materially different from that of the track ACDB, which has the sharp curve CD. Of course the wear on the sharp curve CD per foot of length will be much greater than the wear per foot of length on the track AB, but, on the other hand, the reduction of average track expenses per foot on the straight track AC and DB will cause the general average for track expenses to remain substantially the same. Therefore we may say that when we are compelled to change the course of a line by so many degrees of central angle, it makes no material difference, so far as track expenses are concerned, whether we employ a sharp curve or an easy curve. The sharp curve will concentrate the increase of expenses to within a few feet. The easy curve will merely spread it over a greater distance, but the total extra cost of the curve will be substantially the same in either case.

Fig. 30.

The distinction between the desirability of reducing the rate of curvature in order to attain high speed and the extra, cost of operating freight-trains, and the comparatively low-speed passenger-trains which comprise the business of perhaps 90% of our railroad mileage, must be here clearly appreciated. Although no accuracy is claimed for such a broad statement, it is much more nearly true than any other statement regarding curvature which has equal simplicity.

(2) At what degree of curvature is the total train resistance double its value on a tangent? No one figure will be exact for all conditions. Train resistance varies with the velocity and with the various conditions of train loading even on a tangent, and the ratio of train resistance on a curve and on a tangent varies according to the conditions. As an approximate statement, we may say that a train running at average velocity on a 10° curve will encounter an extra resistance due to curvature which is about equal to the average resistance on a level tangent. We can therefore make a second general statement that on a 10° curve the resistance will be double the resistance on a level tangeat.

(3) The cost of operating a train one mile is approximately so much, say \$I.50. If we double the tractive resistance, we will increase certain items of expenditure, although many other items are unaffected. The combined value of the affected items will aggregate a certain proportion of the cost of a train-mile. A mile of continuous 10° curve contains 528° of central angle, and on the basis of assumption (2) a mile of such track will double the tractive resistance. Therefore, each degree of central angle is responsible for 1/528 of the extra cost of the double tractive resistance. Since the increase as computed is irrespective of the radius and depends only on the number of degrees of central angle, we may therefore say that each degree of central angle of a curve will add that computed percentage to the average operating expense of a train-mile.

This percentage however is based on the extra cost of a curved track over a straight level track. The average figures which we have for the cost of a train-mile are based on the cost of an average mile as it actually exists, including all grades and curves. This cost will evidently be somewhat greater than the cost of operating one mile of straight tangent; but when we consider that the average amount of curvature per mile of track and the average amount of grade per mile of track is quite small, and that its influence in many items in the cost of operating a train-mile is very small, if not zero, we can appreciate the fact that, while the cost of operating a mile of plain level track is far less than that of operating a mile of track with heavy grades and sharp curves, it will not be very much less than the cost of operating a mile of average track. Therefore, although we might make some allowance for this item, we could not allow very much.