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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 III 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.
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.
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 gcd, 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.
Fig. 157. Adjustment of Closed Circuits of Precise Levels.
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 (6 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
Johnson, Theory and Practice of Surveying.
usual way, so that the bubble is at the centre of its run at the instant of reading the staff. When bisecting a bench mark, however, it is convenient to use the elevating screw and note the deviation of the bubble.
Value of one Division of the Bubble Tube.—This is found by sighting on to the levelling staff, whose distance from the instrument is carefully measured. The divisions of the bubble tube are numbered from the centre towards the ends, and the bubble being moved one division at a time, staff readings are taken for each position of the bubble. Readings need only be noted for extreme positions of the bubble, as central and intermediate positions are of little use in finding the mean value of one division.
mean of all the north end readings of the bubble when run to the north end of tube.
Editto, for bubble at south end of tube.
F1 = mean of all the south end readings of the bubble when run to the north end of tube.
F, ditto, for bubble at south end of tube.
$1 mean reading of staff for bubble at north end of tube.
s., ditto, for bubble at south end.
d = distance of staff from instrument.
v = = value of one division of bubble tube (sine of the angle of inclination) at unit distance.
The value of one division of a bubble tube should be constant, but is often altered by changes of temperature of the fastenings of the tube in its case.*
* By experiments on the level of Ramsden's ft. theodolite it was found that though at the ordinary temperature of 66° the value of one division was about 1 second, yet at 32° it was 5 seconds.
Correction for Inclination of Bubble to Upper Surfaces of Rings. This correction is determined in terms of divisions of the bubble tube. It is to be found by reversing the bubble tube on the telescope and taking readings in both positions. An odd number of observations should be made.
= mean of north end readings for bubble tube direct.
reversed. direct. reversed.
Let s1 = staff reading for telescope normal.
where i is the inclination of the bubble tube to the upper surface of the rings in terms of divisions of the bubble tube.
Correction for Collimation Line. This is found by reading the staff with the telescope normal, and then with the telescope inverted, i.c., rotated 180°, about its own axis.
d distance of staff from instrument.
Telescope normal, level normal
where is the correction for unit distance. The correction for any distance is therefore cx distance.
Correction for Inequality in the Size of the Rings. This is determined by reversing the bubble on the rings, and also reversing the telescope in the wyes. The following is an example of the method of ascertaining this correction :—
Reading of Bubble.
Movement of bubble Therefore the bubble moves 2.29 divisions towards the object glass when the telescope is reversed in the wyes. This is evidently twice the difference in the rings, and the angle between the axis of a cone and its slant side being half the apex angle, therefore the line of sight makes an angle with the tops of the rings of one quarter of 2.29 or 0.57 division of the bubble tube. In this case the eye end ring is the smaller, and therefore when the upper surfaces of the rings are level the line of sight is depressed. This correction is termed the "pivot correction," and alters only with unequal wear of the rings.
The angular value of one division of the bubble tube and the inequality of the size of the pivot rings need only be determined once each season. The constant for the stadia hairs which is used for the distances as well as the absolute length of the levelling staff are usually also determined once each season. The inclination of the bubble tube to the upper surfaces of the rings and the collimation line correction are, however, determined daily, at the beginning and at the end of each day's work.
Final Correction.-If D is the difference between the sum of the back sights and the sum of the fore sights, or vice versa, then the final correction is