such a law was in strict conformity with experiments made upon railways and duly reported. The first experiments instituted by me at the time just referred to afforded a complete refutation of this doctrine.

It was found that the accelerawith every increase of speed Thus if a certain speed were

tion was not uniform, but that the acceleration was lessened. gained by a train in one second when moving at five miles an hour, a much less speed was gained in one second when moving ten miles an hour, and a comparatively small speed was gained in the same time when moving at fifteen miles an hour, and so on. In fact, the augmentation of the rate of acceleration appeared to diminish in a very rapid proportion as the speed increased: this suggested to me the probability that a sufficiently great increase of speed would destroy all acceleration, and that the train would at length move at a uniform velocity. In effect, since the moving power which impels a train down an inclined plane of uniform inclination is that fraction of the gross weight of the train which acts in the direction of the plane, this moving power must be necessarily invariable; and as any acceleration which is produced must arise from the excess of this moving power over the resistance opposed to the motion of the train, from whatever causes that resistance may arise, whenever acceleration ceases, the moving force must necessarily be equal to the resistance; and therefore, when a train descends an inclined plane with a uniform velocity, the gross resistance to the motion of the train must be equal to the gross weight of the train resolved in the direction of the plane; or, in other words, it must be equal to that fraction of the whole weight of the train which is expressed by the inclination of the plane. Thus if it be supposed that the plane falls at the rate of one foot in one hundred, then the force impelling the train downwards will be equal to the hundredth part of the weight of the train. So long as the resistance to the motion of the train continues to be less than the hundredth part of its weight, so long will the motion of the train be accelerated; and the more the hundredth part of the weight exceeds the resistance, the more rapid will the acceleration be; and the less the hundredth part of the weight

exceeds the resistance, the less rapid will the acceleration be. If it be true that the amount of resistance increases with the increase of speed, then a speed may at length be attained so great that the amount of resistance to the motion of the train will be equal to the hundredth part of the weight. When that happens, the moving power of a hundredth part of the weight of the train being exactly equal to the resistance to the motion, there is no excess of power to produce acceleration, and therefore the motion of the train will be uniform.

Founded on these principles, a vast number of experiments were made on planes of different inclinations, and with loads of various magnitudes; and it was found, in general, that when a train descended an inclined plane, the rate of acceleration gradually diminished, and at length became uniform; that the uniform speed thus attained depended on the weight, form, and magnitude of the train and the inclination of the plane; that the same train on different inclined planes attained different uniform speeds on the steeper planes a greater speed being attained. From such experiments it followed, contrary to all that had been previously supposed, that the amount of resistance to railway trains had a dependence on the speed; that this dependence was of great practical importance, the resistance being subject to very considerable variation at different speeds, and that this source of resistance arises from the atmosphere which the train encounters. This was rendered obvious by the different amount of resistance to the motion of a train of coaches and to that of a train of low waggons of equal weight.

The former editions of this work having been published before the discovery which has resulted from these experiments, the average amount of resistance to railway trains, there stated, and the conclusions deduced therefrom, were in conformity with what was then known. It was stated

that the resistance to the moving power was practically independent of the speed, and on level rails was at the average rate of about seven pounds and a half per ton. This amount would be equivalent to the gravitation of a load down an inclined plane falling, and consequently in ascending such a plane the moving power would have to encounter twice

the resistance opposed to it on a level. As it was generally assumed that a locomotive-engine could not advantageously vary its tractive power beyond this limit, it was therefore inferred that gradients (as inclinations are called) ought not to be constructed of greater steepness than . It was supposed that in descending gradients more steep than this the train would be accelerated and would require the use of the brake to check its motion, while in ascending such planes the engine would be required to exert more than twice the ordinary tractive power required on level rails. As the resistance produced by the air was not taken into consideration, no distinction was made between heavy trains of goods presenting a frontage and magnitude bearing a small proportion to their gross weight and lighter trains of passenger-coaches presenting great frontage and great magnitude in proportion to their weight. The result of the experiments above explained leads to inferences altogether at variance with those which have been given in former editions of the present work, and which were then universally admitted by railway engineers. The tendency of the results of these experiments show that low gradients on railways are not attended with the advantageous effects which have been hitherto ascribed to them; that, on the contrary, the resistance produced by steeper gradients can be compensated by slackening the speed, so that the power shall be relieved from as much atmospheric resistance by the diminution of velocity as is equal to the increased resistance produced by the gravity of the plane which is ascended. And, on the other hand, in descending the plane the speed may be increased until the resistance produced by the atmosphere is increased to the same amount as that by which the train is relieved of resistance by the declivity down which it moves. Thus, on gradients, the inclination of which is confined within practical limits, the resistance to the moving-power may be preserved uniform, or nearly so, by varying the velocity.

(201.) The series of experiments which have established these general conclusions have not yet been sufficiently extended and varied to supply a correct practical estimate of the limit which it would be most advantageous to impose upon the

gradients of railways; but it is certain that railways may be laid down, without practical disadvantage, with gradients considerably steeper than those to which it has been hitherto the practice to recommend as a limit.

The principle of compensation by varied speed being admitted, it will follow that the time of transit between terminus and terminus of a line of railway laid down with gradients, varying from twenty to thirty feet a mile, will be practically the same as it would be on a line of the same length constructed upon a dead level; and not only will the time of transport be equal, but the quantity of moving power expended will not be materially different. The difference between the circumstances of the transport in the two cases will be merely that, on the undulating line, a varying velocity will be imparted to the train and a varying resistance opposed to the moving power; while on the level line the train would be moved at a uniform speed, and the engine worked against a uniform resistance. These conclusions have been abundantly confirmed by the experiments made in last July with the Hecla engine above referred to. The line of railway between Liverpool and Birmingham on which the experiment was made extended over a distance of ninety-five miles, and the gradients on which the effects were observed varied from a level to thirty feet per mile, a great portion of the line being a dead level. The following table shows the uniform speed with which the train ascended and descended the several gradients, and also the mean of the ascent and descent in each case, as well as the speed upon the level parts of the line:

From this table it is apparent that the gradients do possess the compensating power with respect to speed already mentioned. The discrepancies existing among the mean values of the speed are only what may be fairly ascribed to casual variations in the moving power. The experiment was made under favourable circumstances: little disturbance was produced from the atmosphere; the day was quite calm. In the same experiment it was found that the water evaporated varied very nearly in proportion to the varying resistance, and the amount of that evaporation may be taken as affording an approximation to the mean amount of resistance. Taking the trip to and from Birmingham over the distance of 190 miles, the mean evaporation per mile was 3.36 cubic feet of water. The volume of steam produced by this quantity of water will be determined approximately by calculating the number of revolutions of the driving wheels necessary to move the engine one mile. The driving wheels being 5 feet in diameter, their circumference was 157 feet, and consequently in passing over a mile they would have revolved 336.3 times. Since each revolution consumes four cylinders full of steam, the quantity of steam supplied by the boiler to the cylinders per mile will be found by multiplying the contents of the cylinder by four times 336-3, or 1345-2.

The cylinders of the Hecla were 12 inches diameter, and 18 inches in length, and consequently their contents were 1.28 cubic feet for each cylinder: this being multiplied by 1345.2 gives 1721.86 or 1722 cubic feet of steam per mile. It appears, therefore, that supposing the priming either nothing. or insignificant, which was considered to be the case in these experiments, 3.36 cubic feet of water produced 1722 cubic feet of steam, of the density worked in the cylinders. The ratio, therefore, of the volume of this steam to that of the water producing it, was 1722 to 3.36, or 512-5 to 1. The pressure of steam of this density would be 54.5 pounds per square inch.* Such, therefore, was the limit of the average total pressure of the steam in the cylinders. In this experiment the safetyvalve of the boiler was screwed down to 60 pounds per square

* See Table of Pressures, Temperatures, and Volumes, in appendix.