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For staff readings less than .10 the table does not apply unless the constant (f+d) be added to each of the distances in the second column of the first table. Of course another subsidiary table might be prepared for the hundredths of a foot staff readings, but as .10 corresponds to 12.48 ft. distance readings will not be taken nearer than this, and it is not required.
A table like that given above may be prepared in less than an hour if we remember that omitting the (f+d) the distances are simply proportional to the staff reading.
With such a table and a little practice the field observations of a whole day may be reduced in about fifteen minutes.
Therefore distance strument is 100.
Fig. 183.--Inclined Sights: Staff Vertical.
Inclined Sights: Staff held Vertical. We have hitherto supposed that the collimation line of the telescope is horizontal. In practice, however, it may be inclined at any angle as shown in Fig. 183.
In this case the staff reading ac must be reduced to a'c' at right angles to the collimation line FB. The angle CBC' is equal to the vertical angle v, and the triangle CBC' may be considered to be right angled at c' without appreciable error, therefore we have
= AC COS VS COS V
FB = 100s cos v, if the constant of the inTo this we add (ƒ+d) say 2 ft., and we get―
TB 100S COS V + 2
Also DTB COS V
= 100S COS2 v + 2 cos v
= IOOS COS V sin v + 2 sin v
It is, however, usual to deduce v from D, and we then get
VD tan v
taken as equal to 100, i.e.,
In the above formulæ the constant of the instrument has been
100 and (f + d) = 2. The most
general form of the formula is D
s cos v sin v+(f+d) sin v.
=TB COS V
= 100S COS V + 2 COS V = (100S + 2) COS V For 7 we have v=TB sin v
100s sin v + 2 sin v = (1005 + 2) sin v v is more con conveniently deduced from D, and we have—
= Es cos2 v+(ƒ+d) cos v and
Inclined Sights: Staff held Perpendicular to Line of Sight. In place of holding the staff vertical it may be held perpendicular to the line of sight, as shown in Fig. 184. This is effected by having a sliding sight on the staff and at right angles to it. The staff-holder inclines the staff until he sights the instrument through the sights. The collimation line being then directed to the sight, the staff will be at right angles to the line of sight TB.
In this case if ƒ+d=2, and the constant of the instrument is 100, TB = 1005 + 2, and it is usual to neglect BB'. AS B'GB=V, BBGB sin V, and is usually small.
D is then
BB' = 5 × 1
5-4.35.65 ft. 8 in. nearly error in level.*
2.5 ft. error in distance.
VD tan v.
Errors in Distance and Level with Staff held Perpendicular to Collimation Line.-Taking v=30°, sin v = 1, s= = 6 ft., and distance D therefore about 500 ft., the staff reading at в being 5 ft., we get
* In finding the level of the ground at G the staff reading BG is deducted from the level of the point B or B'; as it is really B'G which should be deducted the error in level of the point G is therefore BG – B'G, as above.
In this case, therefore, it is advisable to take the through levels of the main stations of the traverse with the ordinary spirit level, the leveller following up the tacheometer. The tacheometer levels will be near enough, however, for the spot levels for contours, &c.
Errors of 2.5 ft. in distance and 8 in. in level in a distance of about 500 ft. due merely to the reduction of the sights are very considerable, and to attain anything like accuracy the correction BB' must be calculated and applied to the distance and the staff reading BG must be reduced to B'G, the latter being used in com
Fig. 184.-Inclined Sights: Staff Perpendicular to Line of Sight.
puting the level of the point G. These calculations render the reductions laborious and troublesome, and the system of holding the staff perpendicular to the collimation line is not much used. The system of holding the staff vertical is that most generally adopted.
Error from badly held Staff. The advantage of holding the staff at right angles to the collimation line is that a small error in the position of the staff does not affect the results so much as when the staff is vertical. Thus at a distance of about 600 ft. on the collimation line when v = 30° a deviation of 3° from the vertical position of the staff gives an error of 15 ft. in horizontal
distance, while a deviation of 3° from the perpendicular position of the staff gives only about 1 ft. of error in horizontal distance; for an instrument in which the stadia hairs are so placed that
When using a staff held vertically, it should therefore always be provided with a spirit level to ensure perfect verticality, otherwise the staff-holder must carry a plumb bob. In level country the staff is of course read with the collimation line horizontal, which saves trouble in the reduction of the distance. When possible the lower hair should be directed to an even foot on the staff; this simplifies the subtraction of the lower staff reading from the upper to get s the space intercepted. It is usually more convenient, however, to set the vertical arc to an even minute, so as to get the reductions directly from the tables.
Tables for making the Reductions.—When the staff is held at right angles to the collimation line as in Fig. 184, the reductions may be simply performed with the ordinary tables of sines, cosines, and tangents. When the staff is held vertical, however, to get D we have to multiply by cos2 v and this again by tan v to get v. This would be too tedious without special tables. In making the reductions of inclined sights a table similar to that given on page 269 should be prepared, but without adding in the constant (f+d). The distances as obtained from this table are then to be multiplied by cos2 v and the constant (ƒ+d) cos v added to each. When v=5° 44′ the reduction of the inclined sight to the horizontal is one per cent. exactly, or 1 ft. per 100 ft. Where an error of 1 in 100 is permissible the reduction to the horizontal need not therefore be made for vertical angles less than 6°. For obtaining the elevations of the main stations of the traverse the reduction must, however, be made in all cases if the elevation is deduced from the distance, i.e., from 7 =D tan v.
Stadia Table for Inclined Sights.-Table IV. will be found very convenient for reducing inclined sights in stadia work. This table gives values of cos2 v and cos v sin v for every 2 minutes of vertical angle up to 31°. At the foot of each page values of (f+d) cos v and (ƒ+d) sin v are given for (f+d)=0.75, 1.00, and 1.25.
To reduce any inclined sight it is only necessary to multiply the value of cos2 v or cos v sin v given in the table by the space
274 TACHEOMETRY OR STADIA SURVEYING.
intercepted on the staff and by the constant of the instrument
Reducing the Levels.-Care must be taken in reducing the levels.
For Back Sights we have
When the vertical angle is plus L.I. = L.S.S. + B.S.
For Fore Sights.
When the vertical angle is plus L..S.S. = L.I. + V.c.
These apply whether staff is held vertical or perpendicular to collimation line.
L.I. level of axis of instrument.
L.S.S. level of staff station.
B.S. = back sight (as read by axial hair).
= vertical component =v in formulæ and figures.