them in. Now fix the second straight by a ranging rod or pole near each extremity. Set up the theodolite over the nearest of these points and line in two chaining pins on this line, one on each side of the first line, and as near as possible to it. The intersection point may now be found by stretching a cord line between two of the pins on the first line and another cord between the two pins ranged in on the second line. The required point will then be the intersection of the two cord lines, at which a peg may be driven. The theodolite is then set up over this peg and the intersection angle of the curve measured. Do not forget always to leave a back flag for this purpose at the commencement of the first line, and if there is a sufficient supply of ranging rods, the back flags may be left in and collected by the chainmen as they go out to their work in the morning, thus saving the time required to send a man a long distance back.

Calculation of Length of Tangents. Having measured the intersection angle, now calculate the length of the tangent which is given by

Note that this intersection angle is the angle subtended at the centre of the curve, and that it is 180° minus the angle measured on the ground (see Fig. 172). R should be in chains, then T will be in chains and decimals.

Having the theodolite still set up over the intersection point, now direct the chainmen to measure out the length of the tangent from the intersection point, measuring the second tangent first. This is done with chaining pins, and three pegs are driven at each tangent point, each tangent point peg being carefully lined in with the theodolite.

Shift the theodolite to the first tangent point, leaving a ranging rod at the second tangent point, and proceed to set out the curve. Before doing this the deflection angle for each chain must be calculated, as also the total length of the curve.

Calculation of Deflection Angles.—The deflection angle for chain for the given radius of curve is usually taken from

one of the published tables of deflection angles of curves. If there is not one to hand, it may be calculated as follows :

I

Sin of deflection angle for 1 chain: 2 radius in chains

The deflection angle corresponding to this value of the sine may then be found by looking up a table of sines. This is not strictly correct, although it is generally near enough, as it assumes that the chord of an arc of 1 chain is also I chain in length.

I

Other formulæ are

28.65

Deflection angle for 1 chain in degrees = radius in chains

1719

or deflection angle for 1 chain in minutes

radius in chains

The two last formulæ are arrived at by taking the circular measure of the deflection angle for 1 chain, which is

into degrees; this again multiplied by 60 gives minutes or the second formula.

Exact Formula for Calculation of Deflection Angles.

-The exact deflection angle of any arc =

arc

in circular mea2 radius

sure; this converted into degrees, minutes, and seconds gives the deflection angle to any required degree of accuracy. In accurate work, as for instance curves in tunnels, this formula must be used, and in the case of sharp curves it will also be necessary to calculate the length of the chord for any given arc; thus the chord of a I chain arc on a sharp curve may be 65 ft. 11 in. and an odd fraction of an inch.

Calculation of Length of Curve.-Next to get the length of the curve in chains, divide half the intersection angle by the deflection angle for 1 chain. I For more accurate work reduce the angle subtended at the centre of the curve to circular measure and multiply it by the radius.

Chainage of Tangent Points and Deflection Angle for each Peg on Curve.-Now put in the pegs on the first straight

up to the first tangent point, and measure the distance from the last peg to the tangent point, which gives the chainage of the first tangent point. Add to this the length of the curve and we get the chainage of the second tangent point, i.e., the end of the curve.

The next step is now to note down the deflection angles for each chain round the curve. As the first tangent point or the beginning of the curve is not generally at an even chain, the deflection angle for the first peg on the curve is usually some fraction of the deflection angle for 1 chain. For instance, if the first tangent point is at o miles 25.28 chains, then the distance from it to the first peg on the curve at o miles 26 chains is I chain - .28 chain = .72 chain or 72 links, and the deflection angle for this peg is therefore .72 of the deflection angle for I chain. Add now to this angle the deflection angle for 1 chain and we get the deflection angle for the second peg on the curve, and if we add again the deflection angle for 1 chain we get the deflection angle for the third peg on the curve, and so on to the last peg on the curve.

I

Check on Calculation of Deflection Angles. As a check on the calculation, add to the deflection angle for the last peg the fraction of the deflection angle for 1 chain corresponding to the distance from the last peg to the second tangent point or the end of the curve, and we get the deflection angle for the second tangent point, or in other words, the deflection angle for the whole curve, and this should be equal to half the intersection angle, or at all events within a few seconds of it, the difference being due to neglect of one or two units, as the case may be, in the last decimal place to which the deflection angle for 1 chain has been worked out.

As each deflection angle is calculated it should be entered in the field book opposite the proper chainage, and each angle should be checked off in the field book as it is set off with the theodolite, to avoid confusion.

Laying off the Deflection Angles and Putting in Pegs on Curve. To set out the curve, the theodolite being set up over the first tangent point and the vernier set to zero, direct the cross hairs on to the back flag at the beginning of the first straight, and reverse the telescope and see whether the cross hairs also bisect the intersection point. Now set the vernier to the deflection angle for the whole curve, and see whether the

cross hairs also bisect the second tangent point; if so, the work is so far correct. Now set the vernier to the deflection angle for the first peg on the curve and drive that peg, then lay off the deflection angle for the second peg, and so on until all the pegs have been driven.

Check on Laying off the Deflection Angles.-Now lay off the deflection angle for the whole curve again; the cross hairs should again bisect the second tangent point, otherwise the instrument has been shifted or some slip has been made.

Check on Chaining Round Curve.-Measure now from the last peg to the second tangent point or the end of the curve. This measurement should agree with the chainage of the second tangent point, as calculated from the length of the curve, otherwise the work is not correct, either the chaining of the tangents or the setting out or chaining round the curve being at fault, provided that there is no error in the calculations.

In practice, with fairly good chaining and on ordinary ground, the error in chainage should not be more than half a link on a long curve.

Final Error in Setting out Curve.-When the deflection angle for the whole curve is set off the cross hairs should not be more than 1 or 2 in. at the outside off the second tangent point. In a long curve, in which the theodolite has to be shifted several times, 1 or 2 in. of error is excusable, and may be adjusted by altering slightly the last three or four pegs on the curve.

*

The vernier should now be turned back again to zero, when the cross hairs should again bisect the intersection point and the back flag.

Continuing Work after Setting out Curve. To continue the work, set up the theodolite over the second tangent point, and direct the cross hairs on to the flag ahead on the second straight, turn the telescope over vertically and set off the deflection angle for the whole curve. The cross hairs should now bisect the first tangent point. This is as a check on the tangency of the second straight to the curve set out. The cross hairs are now

* These remarks as regards error apply only to field work of ordinary railways. In the case of curves in tunnels, &c., much greater accuracy is obtained by more accurate measurements, and by adopting special precautions in alignment (see Chapter VII.).

again directed on to the flag at the extremity of the second straight, the pegs on which are lined in at every chain in the same way as on the first straight until the second curve is reached, which is treated in exactly the same way as the first curve already described.

Chaining to be "through."-It is perhaps unnecessary to say that the chaining must be through or continuous from the commencement of the railway to its termination, the tangent points being put in at whatever odd points on the chainage they may occur, but the chain pegs must follow each other consecutively I chain apart whether tangent points intervene or not. In fact, the first peg on the curve may be measured, not from the tangent point, but from the last even chain peg behind it.

The formulæ already given are all that are necessary to set out a curve, but the surveyor should have a thorough knowledge of trigonometry and the properties of circular curves, as many problems may arise, such as inaccessible intersection or tangent points, obstacles in the curve itself, &c. &c. For the many field problems of this nature the reader is referred to one of the various engineers' field books which are devoted to these matters, and which usually contain in addition concise tables of logarithms, logarithmic sines and tangents, natural sines and tangents, deflection angles for curves, &c. &c. Fig. 172 will, however, illustrate the ordinary process of setting out and the principal properties of a circular curve required.

Shifting the Instrument forward to Intermediate Point on Curve. When the curve is long and obstacles intervene so that the second tangent point cannot be seen from the first tangent point, we cannot check the accuracy of the work before beginning to set the pegs on the curve, by laying off the deflection angle for the whole curve to see whether we strike the second tangent point correctly. The theodolite has also to be shifted forward along the curve as the work proceeds, but the deflection angle for the whole curve should always be laid off as soon as the end of the curve is visible, to see that no error has occurred. As a rule, on a long curve the theodolite should be moved forward about every Io chains, otherwise the pegs will get out of line, especially on a sharp curve.

When it is necessary to shift the instrument forward the process