The above problem was purposely chosen with figures in which the pusher grade and the through grade were exactly balanced or equated. When it has once been decided that pusher grades should be used, then the problem of grade reduction is modified to the extent of concentrating effort and attention on the reduction of grades which may be less than the rate of the pusher grade, and to reduce them to the limit of the equated through grade. In other words, if it is decided to retain the 1.9% pusher grade, any intermediate grade between 1.9% and 0.92%, such as 1.4%, must necessarily be treated in one of four ways: (a) it must be reduced to 0.92%; (b) it might be operated as a pusher grade; (c) it might be operated as a through grade, which virtually means that all train-loads are reduced to the 1.4% basis; (d) a fourth possible method would be to use two pusher-engines on the steepest grade, as described below. Of course the first method is the proper method to adopt, if it can be done with a reasonable expenditure of money. The method of pusher grades is only applicable when it is possible to modify the system of grades on the road, so that there is an abrupt change from grades which are to be operated as pusher grades down to through grades which correspond with these pusher grades.

A somewhat unusual solution of a problem in pusher grades is given by the possibility of using two pusher-engines on some grades and one pusher-engine on correspondingly lower grades. Working out such a method on the basis of the engine previously considered, and assuming that the 1.9% grade is the grade for two pusher-engines, we might determine the corresponding grades for one pusher and for through engines as follows:

Tractive power of three engines = 106,000 x 9/40 x 3 = 71,550 pounds. Resistance on 1.9% grade = 6 + (20 x 1.9) = 44 lbs. per ton 71,550 / 44 = 1626 = gross load in tons. 1626- (3 x 107) = 1305 = net load in tons. 1305+(2 x 107) = 1519 = gross load on the one-pusher grade. Tractive power of two engines = 47,700 lbs. 47,70 / 1519 = 31.40 = possible tractive force in lbs. per ton. (31.40 - 6) / 20 = 1.27% = permissible grade for one pusher. 1305+107 = 1412 = gross load on the through grade. Tractive power of one engine = 23,850 lbs. 23,850 / 1412 = 16.89 = possible tractive force in lbs. per ton. (16.89-6) / 20=0.54%=permissible through grade.

On account of its simplicity the above problem has been worked out on the basis that the normal tractive resistance is uniformly six pounds per ton, and also that the normal adhesion of the drivers is 9/40 On the basis of these figures the grades for one, two, and three engines are precisely as given above. Using other types of engines and assuming other values for the resistance and adhesion, the relation of these grades will change somewhat, although, as shown in the following tabular form, the variation will be but slight.

Table XXX. Relation Of One-Pusher And Two-Pusher Grades To Through Grades, With Variations Of The Ratios Of Adhesion And Normal Resistance

Adhesion of drivers.

Resistance per ton.

Load on drivers.

Through grade.

One-pusher grade.

Two-pusher grade.

1/5

6 lbs.

53 tons.

.54%

1.26%

1.86%

1/5

7 "

53 "

.54

1.29

1.92

9/40

6 "

53 "

.54

1.27

1.90

9/40

7 "

53 "

.54

1.31

1.96

1/4

6 "

53 "

.54

1.28

1.93

1/4

7 "

53 "

.54

1.32

2.00

The above form shows that increasing the resistance per ton and decreasing the adhesion have opposite effects on altering the ratios of these grades, and, as a storm would increase the resistance and decrease the adhesion, the changes in the ratio would be compensated, although the absolute reduction in train-load might be considerable. Another practical inaccuracy in the above calculations is obtained from the fact that the rating tonnage on the pusher-engine service is different from that on the through-engine service. To determine the effect on the above case, let us consider the original problem of using a 1.9% grade as a pusher grade using one pusher-engine. Adopting the same figures as before of 2.6 pounds as the resistance of a rating ton and 9 pounds as the resistance of a ton of tare, we have, on the 1.9% grade, a gravity resistance of 38 pounds per ton, and therefore a total resistance of 40.6 pounds per ton for a rating ton. The resistance of a tare ton will be 47 pounds; therefore, to change tare tons into rating tons for a 1.9% grade, we multiply the tare ton by 116% (see Table XXI, § 129). Assuming, as before, an adhesion of 9/40 of the 53 tons on the drivers, we have a tractive force of 23,850 pounds for one engine, or 47,700 for two engines. Dividing 47,700 by 40.6, we have 1175 rating tons for the whole train. The two engines weigh 214 tons, which is the equivalent of 248 rating tons. Subtracting this from 1175, we have 927 rating tons behind the two locomotives. Since the required through grade is still an unknown quantity, we must solve the problem by an assumption of the required through grade in order to determine the equivalent number of rating tons for one locomotive on the unknown grade. We know from the other solution that the through grade will probably be a little less than one per cent, but we know that it will probably be a little higher than the rate given by the other solution, since a locomotive has a higher resistance per ton than the average train resistance. The ratio of tare tons to rating tons on a one per cent grade is 128%, therefore the number of rating tons on the through grade must be very nearly 927 + (107 x 1.28) = 1064 rating tons. Dividing this into 23,850, the tractive power of one locomotive, we have 22.40 the total tractive resistance for one rating ton. Subtracting 2.6, we have 19.8, which is the grade resistance of a 0.99% grade. This value is somewhat higher than the 0.92% grade previously worked out, as was to have been expected. It should be noted, however, that even this value depends on the constants, 2.6 pounds for the resistance of a rating ton and 9 pounds for the resistance of a tare ton, as deduced from Dennis's experiments. If the solution is worked out on the basis of other values, the resuits will probably be somewhat different. On the other hand, it is somewhat encouraging that, in spite of the radical difference in the methods used, the difference in the final results are as small as given above. When separate methods give results which agree to a few hundredths of one per cent in the rate of the grade, we may consider that either are sufficiently accurate for practical use. In view of the variations in train resistance and the uncertainties of the relation between the resistance for a rating ton and the resistance of a tare ton, no table that can be devised will be accurate for all conditions. In Table XXXI is given the corresponding pusher grades for one- and two-pusher, for the same net load behind the locomotive for various through grades. These have been worked out for track resistance of six and eight pounds. The table is valuable in that it affords a very ready comparison of the relative rates of grade under the conditions named. On account of the extra resistance of the extra locomotive, the pusher grades are probably a few hundredths of one per cent too high, and a corresponding allowance should be made. As an illustration of the use of the table, let us answer the question of the permissible pusher grade when the through grade has already been established at 1.24%. If the road-bed is in good condition, we will assume the lower rate of track resistance or six pounds per ton. We will then interpolate between 2.34 and 2.50, and obtain 2.40% as the corresponding pusher grade. For the reason stated above, we should probably cut this down to about 2.3%.

As another illustration, if a road has a few grades of 1.8% which are not of excessive length and yet which cannot be materially reduced except at excessive cost, we may consider the question of operating these few grades as pusher grades, assuming that all through grades can be reduced to the corresponding through-grade limit. Assuming again a track resistance of six pounds and interpolating for 1.8% for one pusher, we have the corresponding through grade of 0.86%. For the same reason as before, this should probably be increased to at least 0.90% to obtain the correct balance. The question then is transformed into the possibility of reducing all grades which are not to be operated as pusher grades, so that none are above the limit of 0.90%. If the road under consideration is already in operation, a closer value may be obtained by considering the actual capabilities of the locomotives employed and the track resistance as it actually exists. For preliminary calculations the above figures are probably sufficiently accurate. It may be noted from Table XXXI that when the track resistance is increased from six to eight pounds per ton, the pusher grade corresponding to any through grade is increased. This is due to the fact that the net load which may be hauled on the through grade is considerably less, so much less that a larger part of the adhesion is available on the pusher grade to overcome grade resistance.