can be utilized for propelling the train is lin amount expressed in Indicated Horse-Power:

ΟΙ

This formula allows 10 p. c. for overcoming the Internal Resistances in the engine itself (see page 11). The friction at the journals of the driving-wheels, however, is not included among these, but is allowed for in the ordinary Rolling Resistance already dealt with. Thus if we take the weight on each driving-wheel as 6 tons, and f= 0.2, the above formula becomes

Thus, e.g., if, in an ordinary locomotive with four drivingwheels, we have the production of steam equivalent to 400 I. H. P., we see that it is unable to utilize its full power for propelling purposes until it attains the speed of about 14 miles per hour, at which point any slight increase in pressure would cause the wheels to slip. Thus up to a certain speed the propelling power of an engine is limited by the weight on its drivers, but remains more or less constant until that speed is attained, after which, instead of being limited by the adhesion of the wheels, it is mainly a question of the steam-producing power of the boiler,

In ordinary practice, 1 square foot of Grate-surface is able, at ordinary speeds, to maintain the production of steam equivalent to 24 I. H. P.: so that if we know the total gratesurface of an engine and the load on its drivers,-assuming it to be tolerably well-proportioned in its various parts,—we can form a fair idea of its tractive power. The usual allowance of grate-surface varies from about 15 square feet in Passenger Engines to double this amount in some of the Heavy Freight Engines thus the power of an ordinary Passenger Engine, when working under ordinary conditions, equals about 360 I. H. P., and in the case of a heavy Freight Engine about 720 I. H. P. Both these classes of engines can, and often do, maintain very much higher powers than these, but to work very considerably above them over a long run is a severe tax on the economy of the engine.

DIAGRAM OF PROPELLING FORCE.

10. In order to ascertain the probable effect of a given locomotive on a certain train on various grades and curves, it is best to draw the Line of Propelling Force of the Engine --i.e., the Line of Tractive Power exerted at the point of contact of the driving-wheels with the rails-in lbs. per ton (2000 lbs.), of the weight of the engine and train.

Suppose, as in Diagram II, we wish to find the effect of a locomotive capable of maintaining a working power of 500 I. H. P. having four drivers with 6 tons on each; and let the engine with its tender weigh 60 tons, and the train be the same as that for which the Lines of Resistance are given in Diagram I, namely, 10 loaded box cars, each weighing 20 tons-f being taken as 0.2. We then have a fair example of the working of a Light Freight Engine.

Draw the Line of Propelling Force as follows:

which (according to Sec. 9) gives the velocity above which slipping cannot occur. Now the theoretic curve of Propelling Force will be a hyperbola, drawn through a (AO and OH being its asymptotes). This curve may be drawn by offsets from OA thus: At a distance along OA from O equal to 10A, the offset equals 4Aa; at a distance equal to OA, the offset equals 2Aa, and so on; the offset varying inversely as its perpendicular distance from 0. Then C, the point of intersection of the Line of Propelling Force with the Line of Resistance, gives the Limiting Speed at which the engine can haul the train, under the conditions for which the line of resistance is drawn,-in this case, on a straight level track.

Then, taking any ordinate such as NMPQ, the part NM included between the Line of Propelling Force and the Line of Resistance gives that portion of the propelling force of the engine in lbs. per ton (2000 lbs.) which goes to overcome the Inertia of the train at the speed indicated.

But this Line of Propelling Force assumes-as we mentioned before---that 10 per cent of the I. H. P. is absorbed in

overcoming the Internal Frictional Resistances of the engine itself-exclusive of the resistance at the journals--independent of the velocity. At low speeds this allowance is considerably too much, but at high velocities it is insufficient; for ordinary speeds, however, it will not be far from correct. The journal-friction forms probably about one third of the whole the friction of the piston, slide-valve, valve-gear, and cross-heads also contribute considerably to the total. Very little is known as to what allowance ought to be made to cover these resistances,—in fact it is so much a matter of lubrication and mechanical detail that no general formula could be applied, but undoubtedly they increase with the velocity, and are higher in an engine hauling a heavy train than in an engine running light.

Also we have Back-pressure of the steam in the cylinders, Wire-drawing, and various other causes entering into the question at high speeds which also tend to lessen the effective Horse-power. -See Note A, Appendix.

11. Now since the loss of power due to these causes depends largely on the rotary velocity of the Driving-wheels, in the case of two engines both developing the same I. H. P. at the same speed,―the cylinders being suitably proportioned, -the engine with the larger wheels will have a great advantage over the other at high speeds, although at low speeds the engine with the smaller wheels will have the best of it. At low speeds-since the initial pressure in the cylinders then differs but little from the boiler-pressure and the back-pressure is practically nothing-an engine with several small drivers will of course have an enormous advantage over an engine of the same I. H. P. with only a single pair of large drivers on account of its being able to utilize so much more of its power, by reason of its higher adhesive qualities. For instance, it would probably tax the engine with large drivers severely to start a train which the other engine could handle with ease; but when the speed reached, say, thirty miles per hour, the engine with the large drivers could work it much more easily and economically than the engine with the small ones. where high velocities are required,-whether on heavy grades or not, provided the weight on the drivers is sufficient,-if the cylinders, etc., are suitably proportioned, the wheels of large diameter are decidedly the best.

Thus

Mr. Wellington states that in the case of ordinary Passenger Engines and trains of medium length, 50 per cent of the I. H. P. is consumed in the locomotive itself, overcoming its various resistances-atmospheric, rolling, internal, etc.,-so that only one half of the Horse-power produced is transmitted through the draw-bar.

From the foregoing it appears that a closer approximation to the true line of propelling force at high velocities may be found by drawing it as shown by the dotted line in Diagram II, somewhat below the theoretic line already drawn. The intersection of this line with OH (produced) gives the maximum speed of the engine if unopposed by any external resistances,―i.e., if running free as a stationary engine,-10 per cent only of the power developed being absorbed in overcoming internal resistances.

It must be remembered that the Line of Propelling Force shown in the Diagram is at all points the maximum which can be obtained without exceeding the I. H. P. stated; but by taking a comparatively low value of ƒ, and a high allowance for the internal frictional resistances of the engine at low speeds, we obtain by the method given probably as correct results as can be obtained by any mathematical process.

12. If we require to know what I. H. P. an Engine must develop to haul a certain train at a given velocity V, we can find it at once theoretically by multiplying the total weight of the engine and train in tons (2000 lbs.) by the resistance in lbs. per ton (taken from Diagram I) and multiplying the product by .003 V (V being in miles per hour). Thus with the train given in Diagram II, we should need an engine capable of developing about 950 I. H. P. in order to haul it at a speed of 50 miles per hour. The I. H. P. exerted increases nearly as V3, and the tractive force nearly as V2. The total amount of steam used theoretically, on a run, is nearly proportional to V2. The most economical speed, as regards fuel, at which a train can be run--provided the engine is of a power suitable to the weight of the train--is found by experiment to be about 18 miles per hour, and not, as might be expected from Diagram I, at about 8 miles per hour. This is due mainly to the saving in heat owing to the engine being a shorter time on the trip, and also on account of the smaller effect produced by variations in grade at the higher

velocity. To ascertain the Limiting Grade which it is possible to work, we find from the diagrams that an engine and tender weighing together 60 tons, with 24 tons on the drivers, can under ordinary conditions just make head-way up a 12per-cent grade; and that it is just all two engines of the above description can do to haul a passenger coach up a 10per-cent grade.

13. The following may be taken as fair examples of the WEIGHT OF AMERICAN ROLLING-STOCK:

RESISTANCE DUE TO INERTIA.

14. We are now able to calculate with a fair amount of precision the Propelling Force of an engine and the Total Resistance opposed to it at any given speed. The Difference between these two, such as is represented by NM, in Diagram II, gives the force in lbs. per ton which goes to overcome the inertia of the train: if the Propelling Force be the greater, increasing the velocity; but if the Resistance be the greater, decreasing it. We will first consider the subject on the assumption that the accelerating force remains constant at all speeds, and that there are no frictional resistances.

It is found by experiment that a force of 1 lb. acting on a weight of 32.2 lbs. (which is perfectly free to move in the direction in which the force is acting) will, after acting on it for 1 second, give it a velocity of 1 foot per second; and that the velocity at all points increases in proportion to the interval of