other portion G C may be found in the same manner as in the problem just referred to; and, having marked the tangental points B, G, C, and determined the radii B O, CO', from the mechanical curves, the operation will be complete.

PROBLEM XI.

1. To find one of the radii of the serpentine curve by calculation, the other radius and tangent points being given. (See figure to last Problem.)

By using the same notation as in Prob. IX., we shall have the following general formula for finding one of the radii of the curve when the other is given, whether the curve be serpentine or compound.

In the present Problem the upper signs are used; and by assigning the proper values to R, &, a, and 8, the length of the required radius r will be found.

NOTE. It may thus be readily known whether the required radius is equal to or greater or less than 80 chains; if less, either the given radius must be, if possible, diminished, or the distance of the tangental points increased according to the requirements of the case. Also, by joining BG, CG; and determining either of the isosceles triangles OBG, O'CG, the position of the common normal point, or point of contrary flexure of the curve, may be exactly found, for the purpose of setting out the curve on the ground.

2. When the two radii B O, C O', of the serpentine curve are equal to find this common radius by calculation, the tangental points being given. (See last figure.) In this case R = fore; also put σ =

r =

r, the other notation being the same as bearc to cos. (cos. a + cos. ß); then,

sine a + sine ß + 2 sine o •†

NOTE. By this very concise formula the common radius may be readily determined; thus giving the most preferable method of forming the serpentine curve, when the nature of the ground will admit of its being done. It thus may be readily ascertained from the data whether the radius be greater, equal to, or less than 80 chains; and, if less, the distance of the tangental points must be increased, till the required length be obtained; but, as already noticed, the limitation of the radii to be either equal to, or greater than, 80 chains, is not in all cases absolutely necessary.

By drawing the Ls Op, O'p', and proceeding with the investigation in the same manner as in foot note to Prob. IX., the general result above referred to, will be obtained.

This formula is a particular case of that given in Prob. IX.; its investigation is also fully given in Baker's Railway Engineering.

PROBLEM XII.

To make a given deviation from a straight line of railway, by three curves; that the works of the line may avoid a building or other obstruction, situated on or near to it.

1. Let ABCD be a straight portion of the railway, h a

building or other obstruction on the line. Take HQ of a sufficient length for a deviation, that the lateral works of the line may avoid the object ath; and through Qdraw a curve GQG of radius QO'= to, or greater than 1 mile. Draw also, two other curves, BG, G'C,

of like radius, to touch the first curve at G and G', and the line at B and C: then the lines OO', O′′O′, joining the centres of the curves, will pass through their points of contrary flexure at G and G'.

2. Calculation. Put r = common radius OB = O'Q = 0′′C, and d = required deviation = HQ; then BH = HC = d(4rd), and the four equal chords B G, GQ, &c., are each = √dr.

EXAMPLE.

=

Let the deviation QH = d = 3 chains, and the radius B O = r = 1 mile = 80 chains; then BH = HC: √3(320—3) = 30.84 chains, and B G = G Q = &c. = √240 = 15.49 chains. These distances being set out will give the required points in the deviation curve BGQG' C, as required.

NOTE. These deviations are now more frequently made than formerly by some of our most eminent engineers to avoid large cuttings and embankments; thus contributing greatly to the economy of construction, especially of branch lines where great velocities of transit are not required.

EXAMPLES OF EXPENSIVE SEVERANCE OF PROPERTY BY
IMPROPERLY SETTING OUT RAILWAY CURVES.

1. It has been already shewn (Prob. I.) that the curve adopted in joining two straight portions of a railway, ought to avoid, as far as possible, expensive cutting, severance, bridges, &c., provided its radius be equal to or greater than one mile, if other considerations do not require it to be less. The following

injudicious violations of this rule came immediately under the author's observation.

In the annexed figure, AR SB is a curve of 11⁄2 miles radius, at Westwick, near Cambridge, in the Wisbeach, St. Ives, and Cambridge Railway: this curve unnecessarily makes an expensive severance, by being laid out through a gentleman's park, with 100 yards of his house, and by crossing and recrossing a brook 16 feet in

width at R and S, within the space of a few chains, the brook being left by the curve only 30 yards, at the widest part between the points of crossing. Thus the

line here requires two oblique bridges at R and S, which might have been avoided, as well as the expensive severance of the park, by extending the tangental portions A C, BD of the railway a few chains each, and substituting a curve CD of one mile radius to supply the place of the one thus ignorantly adopted; the ground being almost perfectly level, and the limit of deviation at the same time, admitting, nay at once suggesting, this improvement of the line, as it respects expense of construction.

2. A similarly expensive severance was unnecessarily made in the same line at Histon, about three miles from Cambridge, by laying out a curve of 3 miles radius, through cottages and gardens, all of which (being situated within three chains of the middle of the line on the north side thereof,) would have to be purchased by the railway company; besides the curve at the same place crossed two public roads within 40 yards of their junction but had the straight portions of the line been extended at both ends of the curve, as in the former case, and a curve of 1 or 11⁄2 miles radius been employed, instead of the 3 mile radius, both the cottages and gardens might have been passed near their southern boundary, and only one road would be required to be crossed, at a short distance from the junction of the two roads in question, the limits of deviation and the ground (being perfectly level) at once suggesting this improvefhent to any engineer of ordinary skill.

There was a slight hope that the head engineer of this line would have rectified these expensive blunders, previous to the formation of the railway: although such important matters were too often overlooked or disregarded in the Hudsonian

hurry of railway projects; which, I understand, was the case on this occasion.

THE PRACTICE OF ENGINEERS IN THE ADOPTION OF CURVES IN VARIOUS RAILWAYS.

The practice of engineers differs very widely in the adoption of curves, some choosing curves of large radius, at great sacri. fices of cost, and others adopting very small ones to avoid expensive cuttings, embankments, &c.,

On the Great Western Railway, "the curves are in general very slight, chiefly of 4, 5, and 6 miles radius. Mr. Brunel considered that even a mile radius is not desirable, except at the entrance of a depôt, where the speed of the engines is always greatly slackened. And, except in these instances, the only deviation from this rule, which he has admitted, is in the curve, about one-fourth of a mile below one of the inclines, where the radius is three-fourths of a mile.”—Railway Magazine, vol. I., page 418.

Mr. R. Stephenson, in his evidence on the projected Brighton Railway in 1836, stated that no curve had a less radius than 11⁄2 miles, in the line he proposed, which he considered a most convenient radius for the high velocities required for passenger trains. Mr. Stephenson does not, however, limit the minimum curvatures to three-fourths of a mile, if engineering difficulties, or other considerations of a sufficiently important character, suggest the adoption of curves of smaller radius.

On the Chester and Birkenhead, Birmingham and Derby, Edinburgh and Glasgow, Arbroath and Forfar, and many others, the minimum radius adopted for the curves is 1 mile; and the same radius is also the minimum in the Birmingham and Gloucester, and the Sheffield and Manchester lines, which are curved throughout almost the whole of their lengths. The London and Birmingham line, though constructed through a very uneven country, has chiefly curves of a radius exceeding 1 mile; while the Manchester and Leeds line has curves generally of three-fourths of a mile radius, with a few considerably less.

A still greater deviation from the minimum limit of one mile for the radius of curves, will be found in railways where great engineering difficulties were to be encountered, especially in mineral lines. The Taff Vale Railway, which is a single line, appears from Sir F. Smith's report to have curves of the subjoined radii and length.

These curves were used to avoid the repeated crossing and recrossing the river Taff, and to avoid the formation of several lofty embankments. The same considerations have led to the adoption of curves of similarly small radius in other mineral lines.

Pambour's formula may be successfully adopted to assign a proper super-elevation to the exterior rail to counteract the centrifugal force, arising from high velocities of trains in curves of small radius. This formula, with the results deducible therefrom, shall be given in the last chapter of this work.

CARELESS EXPENDITURE IN THE CONSTRUCTION OF RAILWAYS, FROM THE NON-ADOPTION OF CURVES, FROM IMPROPERLY LAYING OUT GRADIENTS, EXACTIONS OF LAND-OWNERS, &c. In addition to the expensive and unnecessary severance of property by improperly setting out railway curves, of which examples, are given at page 175, similar wasteful expenditure has resulted from the non-adoption of curves in numerous instances. It has been shewn in Prob. XII. that a lateral deviation may be made in a straight line of railway to avoid buildings or other expensive property, deep cuttings, or other engineering difficulties. This precaution, in a vast majority of instances, has not been adopted, as may be seen in the construction of a great many of the more early railways; and much unnecessary expense has thus been incurred.

Also, where the geological character of the country through which the railway passes, differs considerably, presenting material for excavation throughout the length of the line, varying from loose sand to hard rock, and vice versa; through such varying