of the chain, and moves it to right or left as directed until it is bisected by the cross hairs, when he inserts the rod into the ground and then drives in a peg at the mark thus made. The surveyor will direct the peg to be knocked to right or left until it is bisected by the cross hairs and is correctly in line, and the leading chainman should then apply the end of the chain to the top of the peg and make a scratch on it along the chain handle with a chaining pin. Very often each peg is merely driven so that its centre is at the end of the chain as nearly as may be. In a long line it will be necessary to shift the instrument forward about every 15 chains or so, otherwise the pegs will get out of line, as the ranging rod and the pegs cannot be seen with sufficient distinctness to get them exactly into line at greater distances. In this case, by a signal prearranged with the leading chainman, the surveyor will range in a chaining pin held on the top of the peg, with which a point mark is made on it, and over which the theodolite may be set up. In this work the leading chainman should be the more intelligent of the two.

Chainages and Survey of Detail. The chainage of each road, fence, ditch, &c., intersected should be taken and booked, and any alterations from the 25 in. map in existing fences, new buildings, &c., must be surveyed. The box sextant will sometimes come in useful for the purpose of measuring the angles at which fences, &c., cross.

The chainages of roads, fences, ditches, and survey of alteration to existing detail may be taken by recalling the chainmen prior to moving the theodolite forward, and sending one of them forward with it, then with the assistance of the others the fence chainages, &c., are booked, and any details which require to be surveyed are taken. It is a good rule to take everything required as the work proceeds, as going over the ground again leads to loss of time. When, however, surveys of roads, rivers, &c., for some considerable distance on either side of the railway are required, a special survey of these must be made. One or more assistant surveyors should be a help and ought to expedite the work, but it is quite easy for one man to do all that is required if the time permit.

Bridges. When roads or rivers are crossed on the skew and a bridge is contemplated, the proper angle of skew for the bridge,

which will best suit the ground, should be measured on the ground with the theodolite, and a correct survey of the road or river for some distance on each side of the railway should be made. Levels and cross sections along the road will also be very useful in designing the bridge.

Setting out Curves.*-When the pegs have been driven nearly to the end of the first line, scale off the position of the

tangent point of the first curve as nearly as may be from the 25 in. map on which the line is laid down, and direct the chainman to stop driving pegs when he gets to within a chain or two of where the tangent point will be, and continue the chaining with chaining pins until he gets a chain or two beyond the intersection point, lining in the chaining pins with the theodolite and leaving

For classification of curves by "degree," see Chapter XI.

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

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 1 chain in length.

Other formulæ are

Sin of deflection angle for 1 chain

I

28.65

Deflection angle for 1 chain in degrees = radius in chains

or deflection angle for 1 chain in minutes =

=

1719

radius in chains

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

chain radius in chains, or

I

2 radius in chains

which converts the circular measure

180° and multiplying by 3.1416 into degrees; this again multiplied by 60 gives minutes or the second formula.

Exact Formula for Calculation of Deflection Angles. in circular mea

arc

-The exact deflection angle of any arc = 2 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. 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 I 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 1 chain - .28 chain .72 chain or 72 links, and the deflection angle for this peg is therefore .72 of the deflection angle for 1 chain. Add now to this angle the deflection angle for I 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

=

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