Ex. Let 8200 chains, α = 27°, a′ = 50° : then from a table of nat. sines sin. α = =45399, its cos. = 89101; sin. a'='76604, its cos. = 64279: whence (cos. a + cos. α) = 1 (89101 + 64279) =76690 cos. and 2 sin. of arc to cos. 76690 is 1.28334; therefore 200 79.89 chains, or nearly

r=

=

200

=

=

•45399 + 76604+1.28334 2.50337 80 chains, the radius required.

NOTE. The method of forming the serpentine curve with a common radius is much to be preferred to any other, when the nature of the ground will admit of its being done; and more especially so, when the data, as in the preceding example, will only just give a common radius of 80 chains: for, if the radius of one of the portions of the curve had been taken greater than 80 chains, the other radius would have necessarily been less than 80 chains.

The Continental engineers, in laying out serpentine curves, usually place, if possible, a short straight piece of line between the two members of the curve, to ease the transfer of the train from one direction of curvature to the other.

PROBLEM IX.

To make a given deviation HQ from a straight portion of a line of railway AHD by means of three curves, BG, GQG', G'c, having their radii OB, O'Q, o′′c, all equal, in order that the lateral works of the line may avoid the building, or other obstruction b, which is close to the centre of the straight portion of the line.

1.

Construction.-From the given point H draw HQ = given deviation to AD; on QH prolonged, take Qo'= OP = given radius; with the compasses apply PB = QP twice given radius; draw

=

=

(cos. a+cos, a').

given, whence the comp. of Zo'qo=Zoo'q is known, and may be thus expressed, sin. Zoo'q sin. of arc to cos. Whence oQ=PP'=00' x sin. Zoo'q=2r x sin. of arc to cos. (cos. a+cos. a'), d=BQ=BP+PP'+P'c=r sin. a +r sin. a'+2r sin. of arc to cos. (cos. a+cos. a'), from which δ

Q. E. I.

sin. a + sin. a' + 2 sin. of arc to cos. (cos a+ cos. a')' When the radii are unequal, and one of them, as r=o'c is given, and the other R=BO is required; then o'o=R+r, BP=R sin. a, PO=P'Q=R cos. a, o'Q=R cos. a +rcos. a', and oQ=PP' = √√/o'o2 - o'q2 = √√/(R+r)2 — (R cos. a+r cos, a')2, d=R sin. a +r cos. a' + √(R+r)2 − (R cos. a+r cos a'). By transposing and squaring, and remembering that sin.2+cos.2 = 1, &c., there results d2 – 28 (R sin. a+r sin. a') = 2Rr (1-cos. a cos. a' - sin. a sin. a')=2Rr (1 − cos. a-a')=4Rr sin. a

whence

This formula is used for finding the value of Bo',

Note 2, Problem VII., where the symbols are defined. Q. E. I.

BO to AD; through o' draw o'GO parallel to PB, meeting OB in 0; make CH = HB; and join CQ, QB, the latter cutting oo' in G, and the former cutting o'o" (which

known; and, since the common radius BO is given, the construction

of the curve is obvious.+

Ex. Let the given deviation QH

radius BO

=

85 chains; then BH = HC

√2(340-2) = √676 26 chains, and BG = GQ = &c. = √√2 × 85 = √√/170 = 13.04 chains.

REMARKS ON LAYING OUT THE CURVES IN THE FOUR LAST
PROBLEMS.

Having in the four last problems given various methods of determining the radii and common normals, indicating the positions of the tangent points of the parts of the compound, serpentine, and deviation curves, the method of laying out the curves themselves by Problems II., III., IV., or V., according to circumstances, will be readily seen, recollecting that when junction-points of curves of different radii occur, as cc', first fig. to Prob. VI., to commence the operation afresh, by using the radii and tangents of the respective portions of the curve.

Demonstration.-Because QP, BO are parallel, as are also o'o, PB; O'P=B0= (by const.) qo'= given radius, and o'o=PB=2qo'=2Go'; .. GO'=GO=QO=BO, which are similarly proved to be G'o' G'o"-co". Q. E. D.

✦ Demonstration.-Draw oa perpendicular to BQ, bisecting it in a; then by similar triangles BO: Ba=BQ :: BQ: QH, whence BQ24B0. QH, or BG2 =BQ2 = BO. QH, or BG = √ BO. QH. Also BH2=HC=BQ2 - QH2 = 4BO. QH-QH2 = QH (4BO - QH), or BH HC= √ QH (4B0-QH). Q. E. D.

SECTION IV.

RAILWAY EARTH WORKS.

On setting out the width of ground for a railway.

After the centre stumps of the railway have been put down, which, as before observed, are usually at the distance of one chain apart, the line must next be carefully levelled, and the number of the stumps entered in the Level-Book, in a vertical column; and opposite each number, in a second column, the depth of the cuttings or embankments (see Level-Book, page 441); and in a third column, the computed or horizontal half-width of the surface cuttings as found by Problems I. and II. following; the depths of the cuttings and embankments being estimated from the balance line, which is 2 feet below the line of the rails or gradients, the 2 feet being filled up with gravel to form the way, and the beds for the sleepers of the rails.

The side stumps are next to be put down, which must be placed. two on each side of every centre stump, in a direction perpendicular to the length of the line; or, if the line be curved, in a direction perpendicular to the tangent to the curve at the centre stump: the two interior stumps, i.e., those next to the centre one, to mark the width of the cuttings, and the two exterior ones to mark the ditches of the side fences. The distances of every two of the interior stumps are to be entered in the Level-Book, opposite the number of the centre stump, in the main section, in order to ascertain the quantity of cuttings for the contractor; and the distances of every two of exterior stumps from the centre one are to be similarly entered, to ascertain the quantity of land which will be required from each proprietor for the works of the line; which last may be calculated accurately by the method of equidistant ordinates. (See Problem VI.)

PROBLEM I.

To set out the width when the surface of the ground is laterally on the same level as the intended railway.

From the centre stump, perpendicular to the direction of the line, set out half the bottom-width for the cutting, to which add the width of the side-fence, putting down a stump at each distance; then repeat the operation on the other side of the line.

Ex. If the bottom-width of the railway be 30 feet, and the breadth of one of the side fences be 12 feet, required the widths for cutting and for fences.

302 15 ft. dist. of side-stump from centre for

=

=

To set out the width of cuttings or embankments, when the surface of the ground is laterally level, and at a given height above the level of the intended railway, the ratio of the slopes being given.

the slopes, Mm = AaBb the perpendicular depth, and м the middle stump. Multiply the given depth мm by the ratio of the slopes, to which add half the bottom-width Am, or aM: set out this distance from M to C for the half-width of the cuttings, to which add the width of the side fence for the whole half-width, repeating the same operation on the other side of M.

Ex. If the bottom-width AB or ab be 30 feet, the depth of the cuttings 28 feet, and the ratio of the slopes 1: 1, and the width of one of the side fences 12 feet, required the width of the cuttings, and of the land for the works of the railway.

42 + 15 57 feet = MC MD = width of

=

cuttings; and 57 +12=69 feet = MR = MS = width of land. The

doubles of which are the whole widths.

bottom-width

RC, and the ratio of the slopes

If w

=

= AB, α = depth of cuttings width of one of the side-fences

=

ar +

= MR = MS,

and 2 (ar +

w + f)

=

= w + 2ar + 2ƒ= RS. Construction.-Draw AB = given width 30 feet, perpendicular to which, at its middle point m, draw mм = given depth = 28 feet ; through M parallel to AB draw RMS; through A and B draw Aa, Bb parallel to мm; make ac=bD= 11⁄2 times мm, join AC, BD, and make

RC, DS = width of one of the side-fences = 12 feet. Then ABDC is a cross section of the cuttings, CD the surface-width thereof, and RS the whole surface required for the railway.

By inverting the last figure, so that ABDC may represent the cross-section of an embankment, it will be readily seen that the same method will apply, for setting out its half-widths, MC, MR, as that just given for the cuttings; as the dimensions are the same in both

cases.

NOTE 1. The ratio the depth aa or мm.

of the slopes is the proportion of the horizontal line ca to Thus, when the ratio is 1: 1, ca Aa: when the ratio is 2:1, ca = 2Aa, &c. This ratio depends on the nature of the material through which the cuttings are made; if it is close-jointed hard rock, the ratio is or to 1; if soft, or loose-jointed rock, or strong clay, the ratio is 1 or 1 to 1; if springy ground, or loose sand, the ratio is 2 or 2 to 1.

2. The computed half-widths, in the third column of the Level-Book (following Problem V.), are found by this Problem.

PROBLEM III.

To set out the width of the cuttings, when the surface of the ground is laterally sloping, the height of the centre stump above the level of the intended railway, the ratio of the slopes, and the lateral fall or rise of the ground in a given horizontal distance being given.

AC, BD the slopes, and c'D' a horizontal line passing through M, Mc' = MD' being the computed half-widths.

Having set the level so that by turning the telescope two or three chains' length of the line may be seen, if possible, on both sides of it, place a levelling-staff at M, and another at q, and observe the level readings on them the difference of which will be pq; measure with the tape-line in feet the distance Mq on the slope and the horizontal distance Mp; take the computed half-width from the Level-Book, or find it by Problem II., and multiply it by the distance Mq on the slope, and reserve the product. Add and subtract the product of the difference of the staff-readings, and the ratio of the slopes, to and from the horizontal distance Mp, and reserve the sum and difference; divide the reserved product by the reserved sum for the corrected