rities of the ground. The angle of depression to B is observed, and the vertical arc being clamped in that position, the assistant places the levelling staff successively at a, b, c, &c., the centre of the vane being brought into the line of sight CD, and the heights a a', bb', &c., noted.

b

Section.

B

The instrument is afterwards brought to B, from whence the reciprocal angle of elevation to C is observed as a check on the work, and the telescope afterwards directed upon the staff fixed at the third station E in the line of section. The theodolite being clamped in that position, the same operation is repeated to note the irregularities between B and E. In laying the section down upon paper, a horizontal line being drawn, the angles of elevation and depression can be protracted, and the distances laid down on the inclined lines as they were measured by the chain at the time of the observations being taken. The respective heights of the vane of the staff being then laid off from these points in vertical directions, will give the points a, b, c, &c., marking the outline of the ground.

Levelling with the Theodolite not recommended for
Sections.

As regards this method of levelling for sections, it may be observed that in all cases it is inferior in point of accuracy to levelling with the spirit-level, and that it seldom saves much in point of time. A serious objection to it is the

L

necessity for entrusting the reading of the staff to an assistant, and for making a previous careful inspection of the line to determine the site of each station of the theodolite, in order that the staff, when held on the intervening irregularities, may be long enough to be intersected by the line of sight.

May be used in Exceptional Cases with advantage.

This mode of levelling for sections should not therefore be adopted, except in cases where the line of country, of which a section is required, is intersected by deep and precipitous ravines, or sea cliffs, to cross which much time is consumed when levelling with the spirit-level, owing to the difficulty in fixing the instrument in places suited to

the proper reading of the staff, and owing also to the great number of read ings required in a short horizontal distance. With the theodolite, on the contrary, it is sufficient to place the instrument at the top or bottom of the ravine, and take the differences of level as

above described. The leveller should in such an operation supply the staff-holder with a plummet, and take means of ensuring that the staff is held exactly upright, otherwise his work would be incorrect.

If he hesitate thus to expose the result of his work to the chance of the staff-holder not holding the staff upright, he may cause the staff to be held in directions removed from the vertical so as to be at right angles to the line

of sight. To effect this the staff-holder should slowly

incline the staff backwards

and forwards (that is, to and

from the observer). The lowest reading which can then be obtained from the staff will be when it stands perpendicular to the line of sight, because the perpendicular is the shortest line that can be drawn from a fixed point to a straight line.

Corrections for Curvature and Refraction may be generally neglected in levelling for Sections.

In thus levelling with the theodolite when the distances from station to station are long enough to make the effects of curvature and refraction sensible and of practical importance, they should be taken into account. It rarely happens, however, that such a correction is required, because, within such distances as are adopted in practice, the corrections due to these causes are more minute than the errors caused by the difference of elevation between the axis of the theodolite and that of the vane above the ground, and other disturbing causes. For example, the correction for curvature and refraction combined is,

and as the vane is to be raised or depressed according to signs made by the observer at the instrument, a greater distance than half a mile between station and station is but rarely adopted.

LEVELLING WITH THE SPIRIT-LEVEL.

The spirit-level, with its present improved construction, presents the most accurate means of obtaining a section along a continuous line, or of ascertaining the difference of level between isolated stations.

When practicable, place the Spirit-Level midway between the back and fore sight.

The spirit-level serves to trace a series of lines tangent to a great circle passing through the axis of the instrument, the centre of which circle is the centre of the earth; and if the instrument be placed in the middle of each of these straight lines successively, the difference of level between

the extremities of each line will be obtained without any error arising from curvature or refraction or from imperfect adjustment of the line of collimation as already explained.

a,

The mode of proceeding is thus: The level is fixed at and adjusted by the parallel plate-screws; the difference of reading between the first or "back" station o, and the second or "forward" station 1, is registered in the fieldbook (see form page 154) opposite the distance between the stations; the staff at the station 1 is kept unmoved,

while the instrument is taken forward and fixed at b. From 6 the reading of the staff at 1, which then becomes a "back" station, is registered, as also the reading of the next "forward" station 2; from these data the difference of elevation between, not only the stations 1 and 2 is obtained, but also, by combination, the difference between the extremes O and 2 is given. The same process is continued for any required distance, giving the elevation of each intermediate point, as well as the relative height of the extremes.

In common levelling operations, corrections for curvature and refraction may be neglected even when the instrument is not placed half-way between the staves, because the distance at which the staves can be read is so small as to render their effects inappreciable; at a distance of 500 feet, for instance, the correction due to both causes is only 0.00513 of a foot. Therefore, when the line of collimation is itself properly adjusted, the instrument need not necessarily be placed midway between the back and forward stations when a section of an inclined surface of ground is being taken; but its position may be so chosen that observations can be made each way, with the staves at a considerable distance from each other. When the ground is nearly level, it is better to fix the instrument midway between the staves; but when crossing a valley, the instrument, if properly adjusted, should be placed, for the sake of expedition, nearer the back stations in going down the inclination, and nearer the forward stations in