B, the rating, adding 180 tons as above, is 1,160 and 1,230 tons, respectively, giving 94% for 3.57 miles.

On Hill C, the rating, with 180 tons added, is 1,130 and 1,230 tons, making 92% for 4.41 miles.

Taking the same basis as the author, namely, 4.7 lb. per ton, rate of grade × 20, and weight on drivers, gives:

It will be noted that the author uses the weight on the drivers as the criterion, but the tractive power is not directly as the weight on the drivers, some engines being over-cylindered, or under-cylindered; in the class of engines above mentioned the tractive power is 23.35% of the weight on the drivers.

The writer made a study of several dynamometer tests on Hill C. There is a grade of the same rate, about 1 mile long, near this hill, and a station near its foot, but there is sufficient level grade between this station and the foot of the hill to get a good start.

All the engines of the above class, loaded for Hill C, gained speed on the 1-mile grade, but began to fall below the theoretical speed at a point about 2-1/4 miles from the foot of the hill. This condition occurred when the trains stopped at the station and also when they passed it at a rate of some 16 or 18 miles per hour, the speed becoming less and less as the top of the hill was approached.

The writer concludes that the author might stretch his opinion as to using heavier rates of grade on shorter hills than 10 miles, and indeed his diagram seems to intimate as much, and that, for economical operation, the maximum rate of grade should be reduced after a length of about 2 miles has been reached, and more and more in proportion to the length of the hill, in order that the same rating could be applied all over a division.

This conclusion might be modified by local conditions, such as an important town where cars might be added to or taken from the train.

While it does not seem practicable to the writer to calculate what the reduction of rate of grade should be, a consensus of results of operation on different lengths of grade might give sufficient data to reach some conclusion on the matter.

The American Railway Engineering and Maintenance of Way Association has a Committee on "Railway Economics," which is studying such matters, but so far as the writer knows it has not given this question any consideration.

The writer hopes that the author will follow up this subject, and that other members will join, as a full discussion will no doubt bring some results on a question which seems to be highly important.

John C. Trautwine, Jr., Assoc. Am. Soc. C. E. (by letter).—In his collection of data, Mr. Randolph includes two ancient cases taken from the earliest editions (1872-1883) of Trautwine's "Civil Engineer's Pocket-Book," referring to performances on the Mahanoy and Broad Mountain Railroad (now the Frackville Branch of the Reading) and on the Pennsylvania Railroad, respectively.

In the private notes of John C. Trautwine, Sr., these two cases are recorded as follows:

It will be seen that Mr. Randolph is well within bounds in ascribing to the Mahanoy and Broad Mountain case (his No. 10) a date "certainly prior to 1882," the date being given, in the notes, as 1864; while another entry just below it, for the Pennsylvania Railroad case, is dated 1860.

It also seems, as stated by Mr. Randolph, quite probable that the frictional resistance (6 lb. per 2,000 lb.) assumed by him in the calculation is far below the actual for this Case 10. The small, empty, four-wheel cars weighed only 4,400 lb. each. Furthermore, the "tons," in the Trautwine reports of these experiments, were tons of 2,240 lb. On the other hand, the maximum curvature was 12° 45' (not 14°, as given by the author), and the engine was a tank locomotive, whereas the author has credited it with a 25-ton tender.

After making all corrections, it will be found that, in order to bring the point, for this Case 10, up to the author's curve, instead of his 6 lb. per 2,000 lb., a frictional resistance of 66 lb. per 2,000 lb. would be required, a resistance just equal to the gravity resistance on the 3.3% grade, making a total resistance of 132 lb. per 2,000 lb.

While this 66 lb. per ton is very high, it is perhaps not too high for the known conditions, as above described. For modern rolling stock, Mr. A. K. Shurtleff gives the formula:[D]

where C = weight of car and load, in tons of 2,000 lb. This would give, for 4,400-lb. (2.2-ton) cars, a frictional resistance of 42 lb. per 2,000 lb.; and, on the usual assumption of 0.8 lb. per 2,000 lb. for each degree of curvature, the 12.75° curves of this line would give 10 lb. per ton additional, making a total of 52 lb. per 2,000 lb. over and above grade resistance, under modern conditions.

In the 9th to 17th editions of Trautwine (1885-1900), these early accounts were superseded by numerous later instances, including some of those quoted by the author.

In the 18th and 19th editions (1902-1909) are given data respecting performances on the Catawissa Branch of the Reading (Shamokin Division) in 1898-1901. These give the maximum and minimum loads hauled up a nearly continuous grade of 31.47 ft. per mile (0.59%) from Catawissa to Lofty (34.03 miles) by engines of different classes, with different helpers and without helpers.

Table 2 (in which the writer follows the author in assuming frictional resistance at 4.7 lb. per 2,000 lb.) shows the cases giving the maximum and minimum values of the quantity represented by the ordinates in the author's diagram, namely, "Traction, in percentage of weight on drivers."

It will be seen that the maximum percentage (16.1) is practically identical with that found by the author (16) for grade lengths exceeding 17 miles.

Near the middle of the 34-mile distance there is a stretch of 1.51 miles, on which the average grade is only 5.93 ft. per mile (0.112%), and this stretch divides the remaining distance into two practically continuous grades, 19.39 and 13.13 miles long, respectively; but, as the same loads