Available Working Hours. In clear weather, as a rule, not more than three or four hours a day are available for good work, and at times when the days are very hot and the nights very cold it is not possible to get in an hour's good work in the day. Observing Staff Readings. All three horizontal cross hairs are read, the mean being taken as the staff reading, and the two differences between the axial and extreme hair readings are taken out. The stadia hairs being spaced at as nearly as possible equal distances from the axial hair, these two amounts should agree very nearly, otherwise one or more of the three readings is in error. This is a most valuable check on any serious error in the readings, and should be made in the field for each reading before the level or staff is shifted to the next station. Two levelling staves and two staff holders are employed to save time between back sight and fore sight which would be lost with only one staff. Length of Sights.-Under the most favourable conditions, for the best work the length of the sights should not exceed about 300 ft., or say 5 chains (66 ft. chains). With sights of about this length, in favourable weather an experienced man can level about a mile an hour, or 4 miles a day, of precise levels. The instrument must always be shaded from the sun, both while set up on a station and while being moved between stations. It is essential to do this in order to keep the adjustments anything like constant and the length of the bubble from continually changing. Corrections. If the lengths of the back sights and fore sights are made exactly equal for each set up of the level, all instrumental errors are eliminated. On the other hand, any instrumental error obviously merely applies to the excess of the sum of the lengths of the back sights over the sum of the lengths of the fore sights, or vice versa. The length of each of the sights must therefore be computed from the stadia hair readings, as described in Chapter VI. This is easily done with the aid of a short table, such as that described in Chapter VI. The lengths of the back sights are then added up, as also the lengths of the fore sights, and the difference ascertained. Calling this difference 8, and the correction for instrumental error c, we have where is the average value of the vertical angle for the day, described on page 184. manner. This correction is to be added to or subtracted from the elevation of the finishing point of each day's work, and the levels of any intermediate bench marks are to be corrected in a similar For instance, if the sum of the lengths of the fore sights is in excess, and the vertical angle v is minus or an angle of depression, the sum of the fore sight staff readings is too small, and the resulting elevation of the finishing point of that day's work is too high, and the correction c is therefore to be subtracted. The similar application of the correction when the back sight lengths are in excess and also when the vertical angle v is plus or an angle of elevation will be sufficiently obvious. The difference between the sum of the lengths of the back sights and the sum of the lengths of the fore sights should also be corrected for curvature and refraction, but since the errors due to curvature and refraction tend to neutralise each other, and since 8 is usually so small a quantity, it will generally be found that this correction is unnecessary, if reasonable care has been taken to have the back and fore sights of approximately equal length. The correction for curvature and refraction may, however, be applied when required. For curvature and refraction see page 159, Chapter III. Also, owing to the uncertainty of the amount of the refraction, it is much better to avoid the necessity of applying a correction for it by keeping 8 within small limits. Accuracy and Cost of Precise Spirit Levelling.—On the United States Coast and Geodetic Survey the limit of difference between check lines is 5 millimetres × √2K, where K is the distance in kilometres. On the United States Lake Survey it was 10 millimetres x K, and on the Mississippi River Survey it was 5 millimetres K. These limits are respectively equal to .029, .041, and .021 ft. multiplied by the square root of the distance in miles. Where greater discrepancies occurred, the line of levels ບ * Sin v is used because the distances as found from the stadia hair readings are the distances along the inclined line of sight or collimation line. † Johnson, Theory and Practice of Surveying. was run again. The "probable error" of the mean of several observations may be found as follows: Let d1, do, do, &c., be the differences of the various results from the mean, and let Ed2 be the sum of the squares of these differences, n the number of observations, then the probable error of the mean is When there are only two observations, the formula becomes + }d where d is the difference between the two results. The European International Geodetic Association decided upon the following probable errors in the mean or adopted result : 5 millimetres per kilometre = large. On the United States Coast and Geodetic line of precise levels from Sandy Hook to St Louis, 1,109 miles, the probable error per kilometre of the adopted result was ± 1.2 millimetres. Professor J. B. Johnson ran 670 miles of precise levels on the Mississippi River Survey with a probable error in the mean result of 23.5 millimetres for the whole distance, which is less than 1 in., while the probable error per kilometre was ±.7 millimetre.* In the precise spirit levelling undertaken for the general levelling of France, of 28,700 kilometres the probable error for systematic error was 0.12 to 0.18 millimetre per kilometre, and for accidental error 0.79 millimetre per kilometre. The average closing error of polygons averaging 550 kilometres in length was +60 millimetres. Taking into account the systematic error, the probable error in the difference of level found between Marseilles and Dunkirk does not exceed 60 millimetres. The cost of the levelling of the most precise order was 35 francs per kilometre.† On the United States Geological Survey, using a 20 in. level * Johnson, Theory and Practice of Surveying. + The radius of curvature of bubble tube used for the most precise work was 50 metres; for less degrees of precision the radii of bubble tubes were 30 metres and 20 metres. A system of total reflecting prisms was used to enable the observer to read the staff and note position of bubble simultaneously. with bubbles graduated to 10 seconds of arc, there were 10,968 miles levelled and 1,924 permanent bench marks fixed, at an average cost per lineal mile of 19s. 1d., exclusive of cost of instruments. In 111 closed polygons of an average length of 51 miles each, the maximum discrepancy between duplicate determinations in feet was 0.05 distance in miles, minimum 0.02√ distance in miles, and mean 0.03 ✅ distance in miles. Adjustment of Errors in Closed Circuits of Precise Levels. When a line of levels returns to its starting point the actual error is of course known, as the total difference of elevation should be zero. There are two general methods of applying corrections. One is to distribute the error among the lines composing the circuit proportionally to the length of the lines or to the square root of the length of the lines. The other method is to make the corrections proportional to the computed "probable error" of each line as calculated from the differences between each separate result and the mean result. The corrections are computed by the method of least squares, so that they are the most probable ones, i.e., so that the sum of the squares of the corrections is a minimum. b The latter method is very laborious; and if the closing error is simply distributed so that the error on each line is taken as proportional to the square root of the length of the line, the result is practically identical with that arrived at by the more laborious methods. Experience shows that errors in levelling are more nearly proportional to the square root of the distance than to the distance only. a с Fig. 157. Adjustment of Closed Circuits of Precise Levels. When the levelling includes a number of closed circuits, as in Fig. 157, the closing error of polygon abcg is first distributed among the sides ab, bc, cg, ga. In adjusting the polygon ged, the error is to be distributed between the sides cd and dg only, the error in side ge having already been adjusted. Similarly in correcting gde the error is distributed between de and eg only, and for the polygon gfa its closing error is adjusted in the side fa only. It is necessary to correct first the polygon whose closing error is the greatest, then that with the next greatest error, and so on. American Practice in Precise Spirit Levelling.-On pages 194 to 201 are given the instructions of the Mississippi River Commission for Precise Levelling.* These are the result of many years' experience, and will be a good guide in this class of work. In the United States the levelling staff is called the "rod," and the cross hairs are called "wires" or "threads." The levelling instrument used is of the wye type, similar to Fig. 107, or the Kern level already described. The bubble tube is adjusted to be parallel to the upper surface of the rings on the telescope tube which rest in the wyes, the bubble tube resting on these rings; the collimation line is adjusted to coincide with the axis of the wyes, and any difference in the size of the rings themselves is to be determined. Thus three corrections take the place of the single correction given on page 187. The single correction given on page 187 may also be used for the wye level instead of the above three corrections, as it is obvious that if the bubble tube, rings, and collimation line are all corrected with reference to the same line (the axis of the wyes), the result is the same as correcting for their relative positions to each other, i.e., one single correction for the angle of inclination between bubble tube and collimation line gives the same result; the bubble tube being made level by levelling up with the levelling screws and elevation screw for each sight, this angle is the angle made by the collimation line with a truly horizontal line when the bubble is at the centre of its run, as given on page 184. The methods of determining and applying these separate corrections are, however, given below. It is sometimes the practice to read the staff when the bubble is not exactly in the centre of its run, and a correction is then made to the staff reading for the amount of the deviation of the bubble from the centre of its run. For this purpose the value of one division of the bubble tube, i.e., the vertical angle corresponding to a movement of the bubble of one division, is carefully determined. As a general rule the practice of reading the staff with the bubble deviating from the centre of its run is to be avoided, and the instrument should be carefully levelled up in the |