This evidently is the rate of grade of which the resistance just equals the resistance due to the curve. Unfortunately for the simplicity of our calculations curve resistance is variable. It is greater for very low velocities. It depends somewhat on the detailed construction of the rolling-stock, although fortunately the differences in this respect are not great. When starting a train the curve resistance may amount to two pounds per ton per degree of curve. Such a resistance is equivalent to that encountered on a 0.1% grade. On this account the compensaton for a curve which occurs at a known stopping-place for the heaviest trains should be 0.1% per degree of curve. On the other hand, the compensation required for very fast trains may be as low as 0.02% or 0.03% per degree of curve. But these trains are not the trains which are usually limited by grade. It is the comparatively slow and heavy freight-trains which must be chiefly considered in the study of ruling grades. Therefore from 0.04% to 0.05% must be used as the rate of compensation for average conditions. Even these figures must be considered as only applicable to the ordinary and usual degrees of curvature.
It has been found that the resistance on the excessively sharp curvature used on street-railways or on elevated railroads is far less per degree of curve than the above figures would indicate. This is due to the fact that the actual resistance on a curved track is the sum of a number of resistances, some of which are virtually independent of the rate of curvature. Curves which occur immediately below a known stopping-place for all trains need not be compensated, for the extra resistance of the curve will reduce by that amount the work required from train-brakes in stopping the train. On the other hand, if a curve occurs just above a stopping-place it is a serious matter and should be amply compensated. In either case the down-grade traffic is not affected and therefore need not be considered. Although the suggested rate of compensation (0.04% or 0.05%) is possibly somewhat excessive, it has been recommended on the general principle that it is preferable that the compensation should be somewhat ample in order that it shall be sufficient for all cases. It is quite possible, however, that the excessive rate of compensation might require a steeper grade on the tangents, in order that the desired summit shall be reached in a given horizontal distance. In such cases the rate of compensation should be reduced to 0.035% or even 0.03%. Rules for compensation may therefore be stated as follows:
(1) On the upper side of a stopping-place for the heaviest trains compensate 0.10% per degree of curve.
(2) On the lower side of such a stopping-place do not compensate at all.
(3) Ordinarily compensate about 0.05% per degree of curve.
(4) Reduce this rate to 0.04% or even 0.03% per degree of curve, if the grade on the tangents must be increased in order to reach the required summit.
(5) Reduce the rate somewhat for curvature above 8° or 10°.
(6) Curves on minor grades need not be compensated, unless the minor grade is so heavy that the added resistance of the curve would make the total resistance greater than that of the ruling grade.