of the railway. In this case, the straight line, either horizontally, or having one uniform slope, will be the most advantageous. It is this line which ought to be selected, or the nearest practicable one to it, both horizontally and vertically.

3. It would be a great error to suppose that the line may be lengthened circuitously; because, from that means, by getting easy gradients, the velocity will be much increased, since what is gained in velocity, it is obvious, may be easily lost in greater distance.

4. It was formerly supposed (and the hypothesis has been acted on by many engineers) that the entire line of railway should, as nearly as possible, have one uniform slope, with very good gradients, however circuitously almost the line might be made to obtain them.

5. Now, within certain limits, this is doubtless true, but it requires great care, and more science than falls to the lot of most engineers, to be able to determine these with tolerable accuracy.

6. It has also been a maxim with some engineers, that if a uniform slope is impracticable, or if it requires too great a deviation from the straight or direct line, it is necessary at least to endeavour to rise progressively from one extremity of the line to the other, and never to ascend where the line must descend again.

7. These views in art. 6. have been too frequently advocated and acted upon, to escape the observation of those engineers possessing the requisite information, and their fallacy was not likely to remain long unexposed.

8. Accordingly, Baron Prony de l'Ecole des Ponts et Chaussées, at Paris, reflecting on some of them, pointed out their erroneous nature to M. Navier, who had published a tract, "On the means of comparing the respective advantages of different lines of Railway, and on the use of Locomotive Engines."

9. It is clear that if the traction be increased by gravity, when an engine is impelled up an inclined plane, in proportion to the rate of rise, it will be diminished in the same proportion when it descends, especially when the gradients are very good, never exceeding 1 in 300, and generally much less, in which circumstances they do not require the use of the break.

10. On this principle the loss of velocity in ascending one side of a rising ground will be nearly, but not exactly, as will be after

wards shewn, compensated by the gain in descending the other when the slopes are equal, and some aliquot part of it regulated by the difference, if they are unequal, and this compensation will be the more nearly equal, the better the slopes are and the more perfect our engines become.

11. On some such hints as these from M. Prony, M. Navier's analysis* led him to the result, which is important for the establishment of railways, «That in tracing a line of railway, there is no inconvenience in rising higher to redescend afterwards, as long as that does not make it necessary to extend the limit of the slopes. Thus, for example, several lines uniting two given extreme points, upon which it is admitted that the same locomotive engine draws throughout the same train, will be perceptibly equal in respect to the expense of transit, whatever be the height to which they rise, or from which they descend, if their lengths be equal, and if upon any of these lines the steepest slopes do not surpass 1 in 200; that is, they must not be so steep as to require the use of the break. It appears, then, that especial care should be taken to diminish the length of the line of transit, and to lower the limit of the slopes."

12. It is evident that the minimum or least value of both these should, as far as possible, be combined, otherwise one may be improved at too great an expense of the other, by which means certain loss is undoubtedly sustained. To select the cheapest and most efficient line of railway depends upon the following proposition, which is not very easily solved:-To combine the distance between two given points with the gradients, in such a manner as to produce the maximum effect at the minimum expense. Though this proposition, in general, cannot be solved directly, yet, by attending to the preceding principles, an approximate solution may be obtained sufficiently accurate for all ordinary purposes.

13. In estimating the mean value of the gradients throughout a line, the value of each, with its proper sign, should be multiplied by its length, and the algebraical sum of the products divided by the length of the whole line, including the levels in the same measure, will be the mean gradient.

In this case the signs of the ascents must be positive, and those of the descents negative.†

See Macneill's Translation, page 74, formula at foot of the page. The reasonings of Mr Barlow, &c. will afterwards be considered.

+ In Mr Simm's book on Levelling, the following instances of useless ascents are given from Sir Henry Parnell's treatise on Roads :-" As one instance, amongst others, of the serious injury which the public sustains by this system of road-making, the road between London and Barnet may be mentioned, on

14. If the force of traction obtained in this way on two lines connecting the same two extreme points be inversely as their lengths; or if the product of the length of one line, multiplied by its force of traction, be equal to the product of the length of another line multiplied by its force of traction, the effects of those two lines would be equal, or equal tonnage would, by equivalent locomotive engines, be transported along each line in equal times. This follows from the fact, that if the traction on a unit of the line, such, for example, as one mile, be multiplied by the whole length in miles, the product will be the total traction throughout the line, and it will express the power expended in propelling an engine throughout the whole line. Hence the relative effective powers of two lines of railway may be easily estimated, and their respective advantages and disadvantages readily determined.

15. As the length of a line of railway is one of the elements employed to compute the expense of transit, it is clear it should be as short as convenient, and sound principles will admit. This will also reduce the time of transit, for as Navier remarks, page 8 of Macneill's Translation, " It would be committing a great error to suppose we may lengthen the line because the velocity of transport over it is great. The same principle which rendered the establishment of a railway desirable, in order to obtain a mode of transport quicker than any other, re. quires that the shortest lines be sought after, and even to prefer them when sometimes they appear to be disadvantageous in other respects."

16. In order to ascertain the effects of slopes, experiments have been instituted to determine the amount of tractive force

which the total number of perpendicular feet that a horse must now ascend is upwards of 1300, although Barnet is only 500 feet higher than London; and in going from Barnet to London a horse must ascend 800 feet, although London is 500 feet lower than Barnet." Now, as the distance by which these inconveniences might be avoided is not mentioned, no conclusion can be formed either of the judicious or injudicious formation of this road; but in the following from Mr Telford, that gentleman's abilities and sound judgment are obvious, since he both diminishes the height of his summit level and shortens his distance in the new road across the island of Anglesea, thus :—

necessary to propel a ton of burden on the level plane or horizontal line of a well constructed railway. This, of course, varies a little with the quality of the railway as well as with the construction of the carriages, and depends on the total amount of friction. In general it varies from 8 lb. to 9 lb. per ton, and is therefore very commonly assumed at 81⁄2 lb. per ton, an approximation in the present state of railway carriages not far from the truth. Now, in one ton there are 2240 lb., consequently if 2240 be divided by 8.5 the quotient is 264. From this it is inferred that the traction on the level plane is equal to 1–264th part of the weight drawn. But, by the principles of mechanics, the weight is to the power as the length of an inclined plane is to its height. Now, suppose a waggon enters on an inclined plane, rising at the rate of 20 feet in an English mile of 5280 fect, or 1 foot in 264 feet, it follows that another 8 lb. will be added to that on the level, or that twice the force will be necessary to propel the carriage with its load up this ascent at the same velocity as on the level; that is, if 8 lb. per ton be required to propel a carriage or train of waggons at the rate of 20 miles an hour, it would require double that force of traction, or 17 lb. per ton, to keep up that velocity on a rise of 1 in 264 or 20 feet in a mile. It also follows from the same process of reasoning, that a velocity of 20 miles per hour might be kept up on that inclined plane if the train of waggons contained a part of the load only.* Again, if the rise be only 1 in 2000, it will give an additional tractive force of 1.12 lb., which added to 8.5 lb. gives 9.62 lb. the necessary tractive force up this inclination similarly as before. In this way we arrive at a distinct knowledge of the exact amount of tractive power necessary to propel any load up an inclined plane, whatever may be its rise per mile or inclination. This is the only true purpose to which the principle in question is applicable, though attempts have been made by engineers, not well acquainted with the scientific departments of their profession, to apply it to a very different one.†

This is correctly shewn by Barlow, page 475, &c.

+ The purpose to which we refer here is not a little singular, namely, that 20 feet of rise per mile is equal to one mile of distance, and that one mile of a railway rising 20 feet in a mile is in every respect equal to two miles, on the horizontal plane! Thus, neglecting the distance, the expenses of formation, transit, &c. on the second mile entirely.

17. If the train is moving down the descending plane, then the tractive force necessary on the level plane will be diminished by the effects of gravity to keep up the same velocity on the inclined plane as on the level.

18. If the power employed be constant, then there will be a retardation in ascending the inclined plane, and a corresponding acceleration in descending, which will, in well-constructed railways, whose gradients do not exceed 1 in 300, nearly, though not exactly, counterbalance each other.

The modifications necessary on this account will be considered in a subsequent part of this paper, though the principles already established are sufficient for ordinary purposes, and are quite adequate to expose the 20 feet assumption.

19. On the preceding principles will be compared the relative merits of two assumed lines of railway, in which the values of the respective gradients are given in decimal fractions, as being more convenient for this purpose than vulgar fractions.

Now 0.0217453 divided by 33.73 miles, gives 0.0006447 for the mean effect per mile. If this quotient be multiplied by