Construction of the cross section.-Proceed as in Problem II., excepting that A C must here be drawn parallel to B D.

NOTE 1. The formula for finding the corrected half-width, where there is both a cutting and an enbankment is

wherein a is the width of formation level, and the other symbols the same as in Note 1, Prob. II. See Baker's Railway Engineering.

NOTE 2. To find the distance from M to P, where the cutting and em bank ment meet, use the following proportion;

NOTE 3. When the sloping surface of the cutting passes through the middle point of the formation level, that is, when the points M, m, and P, coincide; then w = w, and the formula in Note 1, becomes

In this case the cutting and embankment are equal.

NOTE 4. By inverting the cross section of this Problem, it will at once be seen that a like calculation will be required, in the case of APC being a cutting, and BPD an embankment.

PROBLEM IV.

To find the surface width of a cutting where the ground is very uneven.

R

D'

Am B

=

=

The annexed figure is a cross section of a cutting, wherein the surface CMD of the ground is very uneven. The method of solving this Problem is by approximation, and will be best shown by an example in numbers.

Let AB= 36 feet, Mm 32, and the ratio of the slopes as 2: 1; then MC' = MD' = (32 × 2) + († × 36) = 82 feet; lay off this distance horizontally from M to d; then d is directly above D'. Observe the difference of level readings at M and d, which in this case is 9 feet; which being multiplied by the ratio of the slopes, that is by 2, gives 18 feet approximate distance Dd; whence MD = Md+dD82 + 18 = 100 feet. Again, place a level staff at D, and the reading will be found to

REZ

OF THE

be 9.7 feet greater than that at M, or 9.7—9 = 0·7 feet greater than at d; therefore the place of the point D requires further correction, which is thus effected; 0·7 × 2: = 1.4 feet second correction; whence MD = 100+ 1.4 = 101.4 feet; which, as the latter correction is small, may be safely assumed to be the true distance of D from M, or the horizontal distance Mq. The method of finding the other corrected half width MC= horizontal distance Mc, is the same as that just given, excepting that the repeated corrections are subtracted from the computed width instead of being added thereto. In this manner the horizontal distance of C from M is found to be 61.8 feet.

Construction of the cross section.-Draw A B D' C', as in the preceding Problems; on C' M D', as a datum line lay off the reduced levels of the several undulations of the surface C M d D of the ground; (see Chap. I., p. 148,) then ABD C is the section required.

NOTE 1. When the differences of the levels at M, d and D are very great, it will require three, four, or more approximations, similar to those just given, to each of the corrected half widths.

NOTE 2. This cross section may be inverted for an embankment, as in the preceding Problems.

LEVEL BOOK.

A level book of the following form is used in setting out the cuttings and embankments of railways.

NOTE. The depths in the second column, are found by calculation. (See Levelling, p. 154,) or by careful measurement from the sections; but the latter method is the less correct of the two. The computed half widths, in the 3rd column, are found by Prob. I.; the corrected half widths, in the 4th and 5th column, by Probs. II., III., and IV., and the widths in the last column are the sums of these in the 4th and 5th plus the breadths of the two side fences of the railway.

PROBLEM V.

To calculate the quantity of land for a projected railway. In preparing the estimates for a projected railway, the required quantity of land is commonly found, without respect to the lateral sloping of the surface of the ground, by taking considerable lengths of regularly rising or falling ground, in one calculation, the depths of the ends of such lengths being measured for the purpose with the vertical scale.

RULE. Find the surface widths, fences included, from the given depths, at each end of the given length, by Prob. I.; multiply their sum by the length in chains, and divide the product by 1320 for the product in acres.

EXAMPLE.

Let the length be 16 chains, the depth of the cross sections at the ends 18 and 58 feet, the width of the formation level 36 feet, the ratio of the slopes as 2 to 1, and the width of the fences 6 feet each; required the area of the surface.

To find the exact quantity of land for a railway.

RULE.—Take the widths, at the end of every chain, from the last column of the level book; add continually together the first and last widths, with twice the sum of all the intermediate widths, and divide the result by 1320 for the content in acres.

EXAMPLE.

Required the content corresponding to the several widths, in the preceding level book.

83.00

148.09

49.44

110.92

391.45

2

782.90

twice sum of intermediate widths, 175.20 first width,

•79868 = Oa. 3r. 8p. the content.

NOTE. It is usual in practice to find the contents of the ground required for the works of a railway from the several proprietors, by measurement from the two-chain plan, prepared for the use of the contractors. Copies are also taken from the plan, showing the positions and extents of ground required from the several proprietors for the works of the railway.

CHAPTER IV.

TUNNELLING.

1. When the depth of a railway cutting reaches 60 feet, and the ground afterwards rises rapidly for a considerable distance, the further progress of the works of the line will be the most economically conducted by making a tunnel: previous to the setting out the earthwork of which, the ground, under which it passes, must be again levelled with great care; and, if the tunnel pass beneath a very high summit, the levelling operation must be checked by the method given in Chap. I., p. 156: for, if there be the most trifling inaccuracy in the section, the gradient or gradients, on which the tunnel is proposed to be formed, will not meet at the points shewn on the section, thus greatly embarrassing the mining operation. See the tunnel, Plate III. 2. If the tunnel be formed on a single straight gradient, the

gradient must be so arranged as to incline to one of its extremities, and in order to prevent the accumulation of water in the tunnel. Strong straight poles must be firmly and perpendicularly fixed on the surface of the ground, in the proposed direction of the tunnel, one of which must be at the summit, from whence the direction of both ends of the tunnel may be observed, that the shafts from the surface to the tunnel may be sunk in the true direction. The shafts are usually sunk at the distance of four or five chains from one another, for the purpose of ventilating the tunnel, as well as for drawing the earth, &c. out of it, and for checking the accuracy of the work, during the mining operation, the depths of the shafts being determined by measurement from the section.

3. If the tunnel be a long one, and in springy ground, it would be advisable, if convenient, to form it upon two gradients, inclining to the extremities of the tunnel, as this arrangement would contribute much to the liberation of the water, during the mining operation, which is usually commenced at both ends of the tunnel at the same time,-but, if the work of the tunnel be required to be also commenced at the bottom of the shafts, as soon as they are sunk, which is sometimes the case, the accumulated water must be drawn from the shafts, or a head-way must be driven from the extremities of the tunnel to the shafts to take off the water.-A head-way being only 5 feet in height and 3 in width, may be more readily formed than the more extensive works of a tunnel, and will be found the most efficient method of draining a tunnel in a wet or springy ground, and thus greatly to facilitate the mining operation.

4. When the tunnel is required to have a curve, through a part or the whole of its direction, the curve must be carefully laid out on the surface of the ground, by one or other of the methods given in Chap. II., proper allowances being made for acclivities and declivities, and strong poles being fixed in the whole direction of the tunnel, as pointed out in Art. (2.), that the shafts may be sunk so as accurately to meet the works of the tunnel below, that their true direction may be secured, this will be more especially necessary where the tunnel is curved, though curves ought, if convenient, to be avoided in tunnels, as accidents, attributable to curves, are more dangerous in tunnels, there being less chance for escape, while assistance cannot be so efficiently given in such secluded situations.

5. If the mining operations of the tunnel be required to commence at the bottoms of the shafts, before the head-ways have been driven to them from the ends of it, the angle that the