point measured down from the collimation line and equal to the various staff readings, then give the points on the surface where the staff was held. Upon joining these points the desired cross section is obtained. With stadia hairs in the telescope the distances along the collimation line may be readily found on the principles described in Chapter VI., without measuring with the tape. For very long cross sections two tapes joined together are required, and on a windy day this is awkward. With the stadia hairs the process is rapid and quite accurate. By having sliding sights at right angles to the staff it may be held at right angles to the collimation line, and in that case the distances along the collimation line may be calculated mentally. In plotting the staff readings are of course then plotted perpendicular to the collimation line. The calculation is troublesome if the staff is held vertical. This method is also useful for cross sections of cuttings on steep hillsides with some rock in the bottom and long earth slopes up the hill. For these it may be necessary to set up the theodolite nearer to the side of the cutting than the centre line, or further out, in order to get the collimation line approximately parallel to the general slope and not too high on the staff. In this case the distance of the instrument from the centre peg is also to be measured and booked and plotted accordingly. The height of the axis of the instrument may be found by levelling with a spirit level a common straight-edge, placed on the centre peg, and measuring from the straight-edge to the axis of the theodolite. The index error of the vertical arc should be ascertained by testing the collimation line on two level pegs 5 or 6 chains apart, and the error is to be added to or deducted from each vertical angle as the case may be.

Contours. In order to fix the position of any point in space we must have its three co-ordinates—that is, its two horizontal co-ordinates and one vertical co-ordinate. The horizontal coordinates fix the position of the point in plan and are determined by surveying, while the vertical co-ordinate fixes the position of the point in elevation and is determined by levelling. The plan ought to show the position of each point both horizontally and vertically, and this is effected by drawing contours on the plan.

Suppose a hill to be cut by a number of equidistant horizontal planes, and suppose we draw the intersections of these

planes with the surface of the hill, as in Fig. 123. Then the lines representing the intersections of the planes with the surface of the hill are 66 contour lines." They are the lines which would be formed by water surrounding the hill and rising a certain height at a time until it reached the top. The edge of the water at each successive rise would form a contour line. It is evident that the steeper the slope the nearer the contour lines will be on the map, while the flatter the slope the further apart the contour lines will be. A right cone would thus be shown by a

Fig. 123.

Contours of a Hill.

Fig. 124.
Right Cone Contours.

Fig. 125. Oblique Cone Contours.

series of concentric circles, as in Fig. 124; an oblique cone, as in Fig. 125.

Vertical Distance between Contour Lines.-This is fixed by the object of the survey, the population and importance of the country, the degree of irregularity of the surface, and the scale of the map. In mountainous districts the contours may be 100 ft. apart vertically. For engineering purposes, such as the general plan of a railway, the contours may be about 5 ft. apart vertically. A good rule is to make the vertical distance between the contours equal to the denominator of the ratio of the scale of the map divided by 600. Thus on the 25 in. Ordnance scale or the vertical distance between the contours would be

25009

2500 ÷ 600, or about 4 ft., which would be about right for locating

a road or railway. On very steep slopes the contours must be further apart vertically, otherwise the contour lines will come too close to each other.

methods.

Determination of Contour Lines.-There are two general The first method is to determine the contours on the ground at once; the second method is to determine the highest and lowest contours only on the ground, and interpolate for the intermediate

15

d

15

15

15

15 b

10

10

10

10

10

10

5

5

5

5

5

locate each contour on the ground with the level, driving in pegs where the line of the contour bends, i.e., at the salient points. The contour lines thus laid out on the ground are then surveyed by any of the ordinary methods, traversing being perhaps best adapted for this purpose. On a long narrow piece of ground, such as that required for locating a road or a railway, take a section across it at every or mile, as at ab, cd, Fig. 126. These sections should be about in the line of steepest slope. Set pegs on these sections at the levels of the desired contour lines. Then the contours are located and marked by pegs by levelling between the corresponding points on ab and cd. Thus contour 10 is located by levelling from 10 on ab to 10 on cd, the levels being checked in between these points. Having thus levelled in pegs at all the points marked a in Fig. 126, these points are then surveyed and plotted on the plan.

Second Method. This method consists in determining the levels and positions of the principal points where the surface changes its slope either in amount or direction. Intermediate contours are then obtained by proportion or interpolation between these points.

The most usual way of effecting this is from cross sections, as described in Chapter V. (see page 226).

Interpolation of Contours.-The levels and positions of

the principal points being fixed, points of any intermediate level corresponding to any required contour are found by pro

portion. Thus in Fig. 127 the position of the points a, b, c, 40.00 35.00 d,e,f,g being as shown, sup

33.00 Fig. 127. -Interpolating Contours.

pose the position of the 45 contour between e and a is required, the distance ae being 100 ft. The total fall between a and e being 15 ft., the distance in which the fall is 5 ft.

will be × 100 = 33 ft., or the 45 contour will be 333 ft. from a towards e. Similarly the positions of any number of contours may be found by interpolation when a certain number of points of known level have been plotted on the plan.

Graphic Method of Interpolating in Contours.-The following method, given by Mr Neil Kennedy, M. Inst. C.E.,* is the simplest and quickest for interpolating in contours. Prepare a diagram similar to that shown in Fig. 128, and trace it on tracing paper or linen. Now suppose that x and y are two points whose levels have been determined, the level of x being 81.00 and the level of y 107.00, and it is required to interpolate in contours every 5 ft. between x and y. Consider the lowest dotted line on the diagram to represent the 80.00 contour, then the tracing paper is moved over the plan until the point a lies on the line of the diagram representing 81.00, while at the same time the point y coincides with the line on the diagram representing 107.00 as shown in Fig. 128; at the same time the diagram must also be so adjusted that the line joining x and y is parallel to the lines a, a, a, which are ruled across the diagram as a guide. If now we prick through the dotted lines of the diagram between x and y we get at once the positions of the 85.00, 90.00, 95.00, 100.00, 105.00 contours between x and y.

When the two points x and y are very near each other, the consecutive radial lines of the diagram may be made to represent contours 5 ft. apart in place of every fifth line of the diagram as in above example.