value of the tractive force is required than this provides for, namely from 4 to 5 tons, the driving-wheels are coupled to one or more pairs of heavily loaded wheels, forming a class of what are called “coupled engines” in contradistinction to the “single engine” with a single pair of loaded driving-wheels. Mechanical energy may be developed in bulk at a central station conveniently situated with regard to a coal-field or a waterfall, and after transformation by means of electric generators into electric energy it may be transmitted to the locomotive and then by means of electric motors be retransformed into mechanical energy at the axles to which the motors are applied. Every axle of an electric locomotive may thus be subjected to a torque, and the large weight which must be put on one pair of wheels in order to secure sufficient adhesion when all the driving is done from one axle may be distributed through as many pairs of wheels as desired. In fact, there need be no specially differentiated locomotive at all. Motors may be applied to every axle in the train, and their individual torques adjusted to values suitable to the weights naturally carried by the several axles. Such an arrangement would be ideally perfect from the point of view of the permanent-way engineer, because it would then be possible to distribute the whole of the load uniformly between the wheels. This perfection of distribution is practically attained in present-day practice by the multiple control system of operating an electric train, where motors are applied to a selected number of axles in the train, all of them being under the perfect control of the driver.

The fundamental difference between the two methods is that while the mechanical energy developed by a steam engine is in the first case applied directly to the driving-axle of the locomotive, in the second case it is transformed into electrical energy, transmitted over relatively long distances, and retransformed into mechanical energy on the driving-axles of the train. In the first case all the driving is done on one or at most two axles; sufficient tractive force being obtained by coupling these axles when necessary to others carrying heavy loads. In the second case every axle in the train may be made a driving-axle if desired, in which case the locomotive as a separate machine disappears. In the second case, however, there are all the losses due to transmission from the central station to the train to be considered, as well as the cost of the transmitting apparatus itself. Ultimately the question resolves itself into one of commercial practicability. For suburban traffic with a service at a few minutes interval and short distances between the stations electric traction has proved itself to be superior in many respects to the steam locomotive, but for main line traffic and long distance runs it has not yet been demonstrated that it is commercially feasible, though it is known to be practically possible. For the methods of electric traction see Traction; the remainder of the present article will be devoted to the steam locomotive.

§ 3. *General Efficiency of Steam Locomotive.*—One pound of good Welsh coal properly burned in the fire-box of a locomotive yields about 15,000 British thermal units of heat at a temperature high enough to enable from 50 to 80% to flow across the boiler-heating surface to the water, the rest escaping up the chimney with the furnace gases. The steam produced in consequence of this heat transference from the furnace gas to the water carries heat to the cylinder, where 7 to 11% is transformed into mechanical energy, the remainder passing away up the chimney with the exhaust steam. The average value of the product of these percentages, namely 0·65 × 0·09 = 0·06 say, may be used to investigate generally the working of a locomotive; the actual value could only be determined by experiment in any particular case. With this assumption, 0·06 is the fraction of the heat energy of the coal which is utilized in the engine cylinders as mechanical work; that is to say, of the 15,000 B.Th.U. produced by the combustion of 1 ℔ of coal, 15,000 × 0·06 = 900 only are available for tractive purposes.

Coals vary much in calorific value, some producing only 12,000 B.Th.U. per ℔ when burnt, whilst 15,500 is obtained from the best Welsh coals. Let E represent the pounds of coal burnt per hour in the fire-box of a locomotive, and let *c* be the calorific value in B.Th.U. per ℔; then the mechanical energy available in foot pounds per hour is approximately 0·06 × 778 × E*c*, and this expressed in horse-power units gives

A “perfect engine” receiving and rejecting steam at the same temperatures as the actual engine of the locomotive, would develop about twice this power, say 1400 I.H.P. This figure represents the ideal but unattainable standard of performance. This question of the standard engine of comparison, and the engine efficiency is considered in § 15 below, and the boiler efficiency in § 11 below.

The indicated horse-power developed by a cylinder may always be ascertained from an indicator diagram and observations of the speed. Let *p* be the mean pressure in pounds per square inch, calculated from an indicator diagram taken from a particular cylinder when the speed of the crank-shaft is *n* revolutions per second. Also let *l* be the length of the stroke in feet and let *a* be the area of one cylinder in square inches, then, assuming two cylinders of equal size,

(8) |

The I.H.P. at any instant is equal to the total rate at which energy is required to overcome the tractive resistance R. The horsepower available at the driving-axle, conveniently called the brake horse-power, is from 20 to 30 % less than the indicated horse-power, and the ratio, B.H.P./I.H.P. = *e*, is called the mechanical efficiency of the steam engine. The relation between the b.h.p. and the torque on the driving-axle is

(9) |

It is usual with steam locomotives to regard the resistance R as including the frictional resistances between the cylinders and the driving-axle, so that the rate at which energy is expended in moving the train is expressed either by the product RV, or by the value of the indicated horse-power, the relation between them being

(10) |

or in terms of the torque

(11) |

The individual factors of the product RV may have any value consistent with equation (10) and with certain practical conditions, so that for a given value of the I.H.P. R must decrease if V increases. Thus if the maximum horse-power which a locomotive can develop is 1000, the tractive resistance R, at 60 m. per hour ( = 88 ft. per second) is R = (1000 × 550)/88 =6250 ℔. If, however, the speed is reduced to 15 m. per hour ( =22 ft. per second) R increases to 25,000 ℔. Thus an engine working at maximum power may be used to haul a relatively light load at a high speed or a heavy load at a slow speed.

§ 4. *Analysis of Train Resistance.*—Train resistance may be analysed into the following components:—

- (1) Journal friction and friction of engine machinery.
- (2) Wind resistance.
- (3) Resistance due to gradients, represented by R
_{g} - (4) Resistance due to miscellaneous causes.
- (5) Resistance due to acceleration, represented by R
_{a}. - (6) Resistance due to curves.

The sum of all these components of resistance is at any instant equal to the resistance represented by R. At a uniform speed on a level straight road 3, 5 and 6 are zero. The total resistance is conveniently divided into two parts: (1) the resistance due to the vehicles hauled by the engine, represented by R_{v}; (2) the resistance of the engine and tender represented by R_{e}. In each of these two cases the resistance can of course be analysed into the six components set out in the above list.

§ 5. *Vehicle Resistance and Draw-bar Pull.*—The power of the engine is applied to the vehicles through the draw-bar, so that the draw-bar pull is a measure of the vehicle resistance. The draw-bar pull for a given load is a function of the speed of the train, and numerous experiments have been made to find the relation connecting the pull with the speed under various conditions. The usual way of experimenting is to put a dynamo meter car (see Dynamometer) between the engine and the train. This car is equipped with apparatus by means of which a continuous record of the draw-bar pull is obtained on a distance base; time indications are also made on the diagram from which the speed at any instant can be deduced. The pull recorded on the diagram includes the resistances due to acceleration and to the gradient on which the train is moving. It is usual to subtract these resistances from the observed pull, so as to obtain the draw-bar pull reduced to what it would be at a uniform speed on the level. This corrected pull is then divided by the weight of the vehicles hauled, in which must be included the weight of the dynamometer car, and the quotient gives the resistance per ton of load hauled at a certain uniform speed on a straight and level road. A series of experiments were made by J. A. F. Aspinall on the Lancashire & Yorkshire railway to ascertain the resistance of trains of bogie passenger carriages of different lengths at varying speeds, and the results are recorded in a paper, “Train Resistance,” *Proc. Inst. C.E.* (1901), vol. 147. Aspinall’s results are expressed by the formula

(12) |

where r_{v}, is the resistance in pounds per ton, S is the speed in miles per hour, and L is the length of the train in feet measured over the