F G E, equal to 60 degrees; and again set off 165 feet to E, from which point measure off the half chord to C; if the flag at C cannot be seen from E, set out the angle 90 degrees (see Diagram); if required, the curve may be continued; sometimes half chord lines may be set out with less trouble than the whole chord.

Problem 27.

Fig. 5, Plate 31. The method adopted in this example differs only in setting out the whole chord by an angle either from the tangent point B, equal to 30 degrees, or by the angle CBA, equal to 150 degrees; the offsets are the same as before. The calculations are made from the radius of 20 chains.

The angle between the tangent and chord is equal to half the angle at the centre in all cases.

If the curve has to be continued, plant the instrument at C, and set off the double angle, equal to 120 degrees.

Problem 28.

Fig. 6, Plate 32. To lay out a curve intercepted by a river, and other obstacles preventing the use of the chain, and substituting two theodolites.

This method has many advantages that is not contained in other examples, particularly on hilly or sloping ground, and where a wide stream of water has to be crossed, as in this instance.

The angles E B C and E C B, as before stated, are regulated by the two radii, and the angle at the centre, that being double the angles E B C or EC B. These angles may be divided into any number of equal parts, as B a, B b, Cd, C c, &c. See the 21st proposition of the third book of Euclid, that all angles contained in the same segment of a circle are equal to one another.

The radius is equal to 20 chains, or 1320 feet.

The angles E B C and ECB are each 30 degrees, which,

for example, is divided into five equal parts, or 6 degrees each.

Plant the theodolites correctly over their respective points B and C; adjust the instruments and set the verniers to zero; then direct both telescopes to E, the point of intersection of the two tangent lines A B and C D; at this point clamp the lower plate, and bring the vernier to the first angle, equal to 6 degrees, whilst the other theodolite will fix the opposite angle at 24 degrees, and so on to the two last points. When the verniers are brought to the respective angles, as at a, b, c, &c., an assistant must be at those points with a pole, waiting the signals from each of the telescopes, shifting the pole until both sights have brought it to the correct point; this operation must be done to each of the points, and a stake driven down.

If it be required to set out each point one chain, or 66 feet apart, the following rule will give the first angle, then the succeeding angles are obtained by multiplying that angle by the number of times, as 1° 26' by 2 is equal to 2° 52', and so on.

Therefore to find the angle so that the points in the curve. shall be 66 feet apart (or any other distance), as by the following rule:

Divide the distance by twice the radius, the natural sine of the quotient will be the required angle, or by logarithms, thus:

Fig. 7, Plate 32, is similar in its operation, having only one theodolite, and the angles are calculated in the same manner. The theodolite is planted at B, and set to the first angle; one end of the chain is held fast at B, the other end at 1 is strained out, and moved about, until by signals from the instrument;

the line of vision intersects the pole held at the end of the chain, which will be the first point; the chain is then moved on and held firm at this point, the other end at 2 waiting the signals from the instrument as before. At all these points stakes must be fixed; so proceed to C, at which point place the theodolite if the curve has to be continued.

This method has similar objections to the first example, except that by this the instrument overcomes difficulties which the chain could not. Problem 30.

Fig. 8, Plate 32. To lay out a compound curve.

This example is introduced to show the method of adopting two curves of different radii, and connect them with the proper tangent points B and C.

The curve B E has a radius of 40 chains, and the angle at the centre 50 degrees; the curve E C has a radius of 80 chains, and the angle at the centre 18 degrees 20 minutes.

As on former examples, the plan should be consulted, and the angles carefully measured, so that the tangent points BE and E C may be minutely defined.

The lengths B a and a C may be correctly obtained by calculation, as shown by Fig. 3; the angle B a O is equal to 64° 30', therefore the angle B a E is equal to 129 degrees, which may be set out by the theodolite, leaving a flag at the point E equal the length of B a.

The offsets are calculated from the tangents as in Fig. 2, and if too long an intermediate tangent should be introduced, as shown in Fig. 3.

The curve EC having a greater radius will require similar calculations, for the tangents Eb and b C, as well as the offsets from them, all of which will come within the limits.

To insure accuracy and prevent extra labour, it is better to fix all the tangents first.

It frequently happens that several curves of different radii are combined, as in the following example.

The centre of every curve must always be fixed on the last radius, as at E O, Fig. 8; and supposing another curve was to join on at C, the centre of it would be on the line C F, and that portion of the line would form the first radius of that

curve.

Problem 31.

Fig. 9, Plate 32. To lay out an inverted curve.

Curves similar to this are frequently adopted, under many peculiar circumstances, to avoid some particular property, or, in an engineering point, to avoid a tunnel, &c. &c., and are sometimes of different radii.

In this example the radii are the same, consequently one calculation answers for the whole, adopting the same rules as before given.

A B and C D show the termination of the main lines, and the respective tangents carried out at a, b, c, d. The centre of the curve E C is fixed in direct position with the line A B, the points BO and Oc forming a perfect parallelogram, and the line O E O dividing it into two equal parts, so that the invert curve B E is the same in every respect as the curve E C; the tangents and offsets will be the same also.

Provided the offsets to the tangents B a, a E b, and b c are too long, the tangents e ƒ and g h must be introduced, then the whole of the tangents will be the same, and require but one calculation.

By referring to the diagram, it will be seen the curve c C has the same radius; but its connexion with the tangent line C D having a different angle, changes the tangents at c d and d C, and would require a fresh calculation.

The preceding examples are carefully selected to meet the many difficulties or obstructions that are likely to occur in setting out railway curves. It must rest with the judgment of the surveyor as to the particular method most suitable to adopt.

Too much care cannot be exercised in the operations. It must be borne in mind, that however careful the calculations

and observations are made, the plan being a perfect level surface, some of the most easy diagrams may have the roughest ground; therefore it is expedient that the tangent lines and tangent points be fixed with minuteness, attending particularly to former instructions in chaining, and allowing the hypothenusal difference.

To find the length of a curve, see Problem 65, Part I., and Table, No. 16.

As Table No. 6 contains the first offsets to tangents already calculated, it will take but a small portion of time in the office to prepare a sufficient number of calculations for the day's operation; and although there are very many useful tables published for setting out railway curves, if the preceding examples are carefully attended to it will render those tables unnecessary.

ON SETTING OUT THE WIDTH OF LAND REQUIRED

FOR A RAILWAY.

The previous observations were chiefly confined in directions to lay out the centre line of the railway; that being done, it is then necessary to level the line again very carefully, taking the heights from off each stake, which are then numbered consecutively throughout, corresponding to the mileage on the working section, which are plotted from these revised levels.

Another book is then prepared from the above, in which are inserted the heights or depths of the cuttings and embankments, and the corrected half widths, as will be hereafter explained.

When the ground is level there is no difficulty in setting out the widths from the centre line, as both sides are equal.

When the ground is higher on one side of the line than the other, the widths from the centre are then unequal, therefore in setting out the width for an embankment or cutting, not only has their slopes to be considered, but the slope of the natural ground also.