angles that these distances make with the tangents to the curve, and give the requisite formula.

XXIV. The widths of a laterally sloping railway cutting from the centre of the line are expressed generally by the following formula.

the positive sign being used for the width measure down the slope, and the negative one for that up the slope, in which formula bwidth of the cutting, assuming its surface to be level, h = differences of level readings at the distances s and l on the slope and level, and r the ratio of the slopes. See figure to Prob. II., Chap. III.

XXV. Let a and b be the depth of a railway cutting to the intersection of the slopes, length of the cutting, w = bottom width, all in feet, and r = ratio of the slopes, the surface of the cutting being assumed to be horizontal; then prove that the content of the cutting in cubic yards is

XXVI. When a and b are the depths of a horizontal cutting to the formation level, and the other dimensions the same as in the last Problem; then prove that the content of the cutting in cubic yards is

XXVII. Let A and B be the areas of the cross sections of a cutting to the intersection of the slopes, d, the area of the end of the prism below the formation level, (see fig. p. 194) and 7 the length of the cutting; then prove that the content in cubic yards is

XXVIII. Let the dimensions be as in Problem XXVI.; then prove that the error in defect of the method of finding the content of a cutting by using the mean depths is

XXIX. Let the dimensions be as in Problem XXVII.;

then prove that the error in excess of the method of finding the content of a cutting by using mean areas is

XXX. Prove the formula at p. 202, for finding the error per cent. of Mr. Bashforth's method of finding the contents of railway cuttings from sectional areas.

NOTE. The demonstrations of the nine last Problems are given in Baker's Railway Engineering.

CENTRIFUGAL FORCE OF TRAINS IN RAILWAY CURVES.

Since all moving bodies have a tendency to continue their motion in a direct line, from this cause the carriages of a railway train of great velocity are strongly urged towards the outer rail, and would ultimately be driven off the rails, were it not for the flanges of the wheels and the conical inclination of their tire.

F=

Let F centrifugal force thus generated, W = weight of the train, V = its velocity, R = radius of the curve, in which the train moves, and g the force of gravity at the earth's surface; then by Dynamics

W V2 9 R

EXAMPLE.

1. When Ra mile = 2640 feet, V = velocity

per hour 44 feet per second, and g

= 30 miles

= 32 feet = velocity of

a body falling from rest, at the end of a second; then

that is, the force that urges the train to quit the curve is of

its whole weight, in this case.

2. When V

60 miles per hour 88 feet per second, and

R the same as in Example 1; then

F=

W x 882

32 × 2640

= nearly W,*

*This great amount of centrifugal force, in curves of small radius, would be very much increased by the high velocities, which some are sanguine enough to expect as likely to be attained on railways; since this force varies as

that is the force, in this case, is of the weight of the train. Hence it may be perceived how extremely dangerous high velocities are in curves of small radius.

3. When the radius is = 1 mile = as in example 2; then

F=W.

5280 feet, and V the same

This force, except in curves of very small radius, is counteracted by the conical inclination of the tire of the wheels of the engine and its train. The inclination with the lateral play of the flanges of the 2 wheels of about an inch on each side, and the centrifugal force urging the train towards the outer rail, when moving in a curve, increase the diameter of the outer wheel and diminish that of the inner one, which causes the train to roll on a conical surface, thus necessarily producing a centripetal force to counteract the tendency of the train to leave the curve. However, in curves of very small radius, the centripetal force thus generated, does not sufficiently counteract the centrifugal force, a proper super-elevation of the exterior or outer rail being required for this purpose; for determining which Pambour has given in his work on Locomotive Engines, the following.

FORMULE FOR THE SUPER-ELEVATION OF THE EXTERIOR RAIL. Let V = velocity of the train, R = radius of railway curve, R' = radius of the curve that the train would describe in consequence of conical shape of the tire of the wheels, and the centrifugal force impelling the train outward, and enlarging the diameter of the outer, and diminish that of the inner wheel, gauge of rails, g force of gravity, and a super-elevation of outer or exterior rail; then

e

=

x=

for the same curve: thus for a velocity of 120 miles per hour, on a curve of a mile radius, we shall have

that is, the centrifugal force is, in this case, more than of the whole weight of the train; while for curves of 1 mile radius, which are very common in railways, fW, or nearly of the weight of the train. It must, therefore be evident that a velocity of 120 miles per hour, or even one of 90 miles per hour, must be extremely dangerous, especially on an embanked curve, should any accident throw the train off the line, which is often the case with the present velocities. Moreover, the resistance of the air, which varies as V2, must be considerably augmented by high winds opposed to the direction of a train of these great velocities; while its engine would require a power greatly superior to those now in use.

d being = outer diameter of the wheels, ▲ = deviation of the

Concise and, he trusts, clear demonstrations of the above formulæ are given by the author, in his Railway Engineering. By solving these formulæ for some of the usual cases, Pambour produces the following.

TABLE OF THE SUPER-ELEVATION TO BE GIVEN TO THE EXTERIOR RAIL IN CURVES.

The correctness of the above results is pretty generally conceded. It must, however, be considered, that it is extremely difficult, if not impossible, to realize in practice, the precise conditions and proportions determined by these important formula; as accidental depressions and enlargements of guage of part of the rails, as well as many other matters that cannot be subjected to calculation, will unavoidably derange these results.

The reader, who wishes for further information on these subjects, may consult Tredgold on the Steam Engine; Hann's Treatise on the Steam Engine; and Baker's Statics and Dynamics.

APPENDIX.

RAILWAY VIADUCTS OR BRIDGES.

The extensive lines of railway lately constructed, and now in progress, present numerous viaducts, some of which are stupendous as well as magnificent works, and some of a truly novel character. The dimensions of the chief specimens are here given.

TUBULAR AND OTHER IRON GIRDER BRIDGES.

The Britannia tubular bridge.-This structure, combining unparallelled magnitude, strength, and novelty, forms one of the viaducts of the Chester and Holyhead Railway. It crosses the Menai Straits, uniting the shores of the mainland of Wales and the Isle of Anglesea. It consists of two rectangular tubes, each 1513 feet in length, or of a mile, 26 feet in average depth, and 11 feet 8 inches in width, the internal depth and width being respectively reduced by the construction to 22 and 14 feet. Each tube has four spans, and consequently three piers or towers, exclusive of the abutments. The two middle

spans, in each tube, are each 460 feet, and the two end spans each 230 feet, exclusive of the widths of the towers, which support the tubes at a height of 102 feet above high water mark, the whole height of the middle tower being 200 feet above high water mark, or 230 feet from its foundation. The parts of the tubes forming the middle spans were 472 feet in length, previous to their being united, and weighed upwards of 1600 tons, and were raised to their present lofty position by hydraulic presses worked by steam engines, thus leaving the navigation of the Menai Straits uninterrupted.

"It is seldom," says Mr. G. D. Dempsey, "that the invention. of works of new design and skilful mechanical arrangement is due entirely to one mind, any more than their construction is due to one pair of hands: hence great difficulty arises in assigning to each contributor his fair share of merit in their production. It must, however, be admitted, that to Mr. R. Stephenson alone we are in this instance indebted for the original suggestion; and with this admission, we have endeavoured to avoid any attempt to judge of the precise claims of the two eminent men, whose joint labours have produced the