Measure also the height of the axis of the instrument above the peg or ground at B.* * The height of the axis of the instrument as deduced from the level of B + height of instrument above ground or peg should agree with height of axis of instrument as deduced from vertical angle and staff readings observed in the back sight from B to A. The distance AB should also agree with the distance as deduced from the fore sight A to B. If these do not agree, the means may be adopted for the plotting in the office. Now all the lateral and intermediate points visible from в are to be observed and the readings booked. Any very important points previously observed from A may also be observed from в as a check and the mean values adopted. Next the station c is observed, and the readings booked. Then shift to c and take a back sight to B, then take all detail visible from c, and so on, until the whole route has been traversed. Special side stations may be used when it is desired to see over an area not visible from the main traverse. In case of any obstacle to reading of extreme stadia hairs or in case of distances so great that both extreme hairs cannot be brought on the staff, use one extreme hair and the axial one and deduce the other extreme hair reading.

Micrometer for Long Sights. When the length of the sights exceeds the power of the instrument to distinguish the figures and divisions of the staff, the micrometer must be used. This consists of an extra pair of movable hairs actuated by a micrometer screw near the eye end of the telescope. These movable hairs are moved apart or closer together until they coincide with the extremities of two discs on the levelling staff or on a special staff. The distance apart of the discs is usually Io ft. From the micrometer reading the distance is deduced. As a rule the working limit of the length of the sights will be about Io chains, beyond which a micrometer should be used.

Best Class of Instrument for Tacheometry. As the ordinary 20-power telescope will not read .or ft. on the staff or to 1 ft. of distance at 10 chains, a 40-power telescope should be stipulated for in the theodolite, either with ordinary stadia

*It is a good plan to have the extremity of the plumb-bob chain exactly 2 ft. below the axis.

T

hairs to read 1 per cent. or 1 ft. per 100 ft., or with an extra anallatic lens.

Office Work.-The first work to be done is to reduce the distances and levels in the field book. When the staff is held at right angles to the collimation line, ordinary tables of sines, cosines, and tangents will suffice. Crelle's tables, which give at one inspection the products of three figures by three figures, will also be of assistance. With the staff held vertical, Table IV. will be useful. The traverse lines are first plotted in the usual manner. The details are next to be plotted by means of radiating lines from each of the traverse stations. Each of these radiating lines will be the bearing of the point where the staff was held. The distances are scaled off, and the reduced levels of the points figured on the plan alongside each. From these reduced levels all the contours are drawn in by interpolating between them. See graphic interpolation of contours, Chapter III. From the contours the proper location of the line is laid down in the usual way (see Chapter V.).

Special Protractor.-In order to avoid unnecessary ruling of radiating lines at each main station and consequent disfigurement of the plan, a special protractor has been devised. It is semicircular in shape, the scale of the plan being marked along the line of its diameter. It rotates round a needle passing through a hole in the protractor and through the main station. The protractor being set with the bearing of the line in coincidence with the meridian line through the main station, its zero will then be on the bearing of the line-that is, the diameter of the protractor will coincide with the position of the line desired to be protracted. The point may therefore be protracted by simply scaling off the distance along the scale on the protractor, after it has been set to the proper bearing.

Accuracy of Tacheometer or Stadia Surveying.When through levels of the main stations are taken with the spirit level, the resulting longitudinal section of the line plotted from the contours will agree very closely with the actual working section. Cross sections may be plotted from the contours, and the quantities of earthwork will also agree closely with the final quantities.

When through levels are taken with the ordinary spirit level the level of the axis of the instrument is of course deduced from these levels + height of axis above peg or ground, and not from the back or fore sights.

There can be no doubt of the rapidity of tacheometry in the field work, while much more detail is likely to be surveyed with the tacheometer than by any of the older methods of surveying. Its use is, however, of course confined to open country. On the Continent the tacheometer is supplanting every other method of making preliminary surveys for public works, especially among Italian, French, German, and Spanish engineers.

Every engineer should at least stipulate for stadia hairs in the telescope of his theodolite.

Mr R. E. Middleton, M. Inst. C.E.,* in order to get practical data of the accuracy of tacheometric distances and levels, measured a base on a plot of ground near Wimbledon, the probable error of which was under in. He had a number of stations set out trigonometrically, the lengths of the lines being calculated from the measured base, while the levels were determined by levelling with the spirit level. All the stations were then surveyed and levelled by tacheometry with a 5 in. theodolite divided to 20'. Two systems were employed, the first being the tangential system by simple vertical angles, described on page 256, and for which the Bell-Elliott tacheometer (Fig. 180) is specially adapted; the second being the tacheometer system proper with stadia hair readings of an ordinary levelling staff divided into feet and hundredths of a foot. The latter is called the subtense system by Mr Middleton. On the back of the staff black lines 13 ft. wide and 10 ft. apart were painted as sights for the tangential system. In the observations of the tangential angles both faces of the instrument were used, two verniers being read at each observation, thus making four readings for each angle. The observations were made by an assistant, so as to represent the accuracy attainable in practice. Observations on the tangential system were also taken to two lines only 5 ft. apart on the staff, and also on the tacheometer system proper, with the interceptions of only two stadia hairs in place of three. The results of the 10 ft. base and three stadia hairs only are given here :—

* Practical Observations in Tacheometry, Minutes Proc. Inst. C.E., vol. cxvi.

The following are the conclusions arrived at by Mr Middleton. It may be seen that the shortest lines, taken separately, are more accurate than the longer lines. The short lines are not, however, so accurate as the long lines when taken in the aggregate and proportionally.

Comparison of the two Systems. A comparison shows that the limit of accuracy is reached in the tangential system when the staff is 1,000 ft. from the instrument; as in lines between 100 and 1,000 ft. long the average error per 1,000 ft. is +0.116 ft., These figures are arrived at by dividing the totals of the errors by the number of lines.

+ These figures are arrived at by dividing the final error by the number of lines, and reducing to a length of 1,000 ft. in simple proportion.

while in lines above 1,000 ft. long the average error is +2.44 ft. In the subtense system the limit of accuracy appears to be reached at 800 ft. Between 100 and 800 ft. the average error is -0.43 ft. per 1,000 ft., while beyond that distance it increases to −1.97 ft. (see Tables). Within the above limits the results are decidedly in favour of the subtense system as regards individual errors, and are not much affected by distance.

The degree of error of the subtense system may be represented by a probable error of ± 1 ft. per 2,000 ft., which compares very favourably with careful chaining, and is much superior to chaining on rough ground. As regards the reducing, there is not much difference between the labour involved, but with proper tables the advantage would probably lie with the subtense system. As regards manipulation there does not appear to be much difference between the two systems, as each is easily mastered. For greater lengths than 1,000 ft. in the one case and 800 ft. in the other the tangential system is best, especially if sighting discs are attached to the staff. With these by the tangential system distances up to a mile may be measured with some degree of accuracy.

In using the tacheometer, the notion of English engineers that the instrument is the responsible post must be abandoned, and the chief must devote himself to directing the staff-holders and recording the positions where the staff is held, leaving the manipulation of the instrument and the recording of the observations in the field book to assistants.

Levelling. In levelling with the theodolite the two systems are identical, and the results attained the same. In 160 continuous observations, the average error per observation was +0.152 ft. The closing error in a length of 7,247 ft. was ±0.54 ft.; in a length of 23,744 ft. it was ±0.65 ft. This makes the probable error per mile 0.40 ft. in one case and 0.30 ft. in the other. If the error of each station be examined independently, and the results averaged, the error in 160 observations, the total length of the lines being 92,269 ft., and the average length of each line 576 ft., is at the rate of 0.0067 ft. per observation, and 0.02 ft. per mile.

Of lines between o and 800 ft. in length, the number is 131, with an average length of 436 ft., the error per observation being 0.017 ft., and the probable error per mile 0.06 ft.