corresponds to the angle twice the tangent of half of which is 0.02, while the measuring angle formed by the other set corresponds to that angle twice the tangent of half of which is 0.004. When the line of sight of the furthest apart set is obscured the other set is to be used, the staff reading being multiplied by 5 as .004 x 5 = .02; the .02 set being as a rule used. The eyepiece can be moved up and down by the pinion shown so as to obtain a clear sight of the extreme cross hairs. The bubble at A is connected to the vernier arms of the vertical circle. For this instrument the size of the divisions on the staff must be .02 of the unit of measurement, i.e., .02 ft. or .02 metre as the case may be. For all practical purposes, however, an ordinary theodolite, having cross hairs fitted so that staff are also .01, i.e., .01 ft. or .01 metre, which is the ordinary division of the levelling staff, is all that is necessary for tacheometry work. Specially Graduated Staff.-A staff may be graduated to suit the stadia hairs of any particular theodolite, so that if the constant multiplier = 103.20 say, using an ordinary staff graduated to feet and hundredths, the multiplier is exactly 100 using the specially graduated staff. This is of course effected by making the size of the divisions of the special staff to suit the instrument in question. This system has been found less accurate than the system which uses the ordinary levelling staff graduated to feet and hundredths, and is objectionable for the following reasons :—' -The cost and trouble of regraduating and repainting the staff when the constant of the instrument changes, as when new stadia hairs are put in or for other reasons; the staff cannot be used with any other instrument except that for which it is made; such a staff could not be used for obtaining the levels of the various points observed without calculating laborious corrections. Stadia Tables for Horizontal Sights.-Many engineers have objected to using the stadia system on account of the labour involved in reducing the field observations. It is, however, easy to prepare a table for any instrument by means of which the labour of reducing the field observations is much diminished. Suppose for the theodolite in use = 104.80, ƒ+d=2 ft., such a table would be as under : Suppose the staff reading is 4.73, from the table we have That is, the distance corresponding to a staff reading of 4.73 is 497.70. This is obtained from the table by inspection and the simple addition of three quantities. In preparing the table the constant (ƒ+d) =2.00 is to be included in the distances for the even feet staff readings only, i.e., in the last column only. If the (ƒ+d) were included in the distances for the tenths and hundredths of a foot also the result would be that the (ƒ+d) would be added in twice or three times in place of once only, in taking out any distance from the table. For staff readings less than 1 ft. the subsidiary table for tenths given under the principal table is to be used. This is simply the distances given in the fourth column of the first table with the constant (f+d) added to each. The parts for the hundredths of a foot are taken from the first table as before. For example, for a staff reading of 0.99 we have 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. 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 right angles to the collimation line FB. must be reduced to a'c' at 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 Therefore distance strument is 100. 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 It is, however, usual to deduce v from D, and we then get— V=D tan v (1) (2) In the above formulæ the constant of the instrument has been f The most taken as equal to 100, i.e., { = 100 and (f + d) = 2. general form of the formulæ is D = cos2 v+(f+d) cos v and V = cos v sin v + (f+d) sin v. 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, BB': = GB sin v, and is usually small. D is then TB COS V = 100S COS V + 2 COS V = (100S + 2) cos V. For v we have v = TB sin v = 100s sin v + 2 sin v = (1005 + 2) sin v v is more conveniently deduced from D, and we have (3) (4) 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 BB'=5×2.5 ft. error in distance. 5-4.35.65 ft. 8 in. nearly error in level.* * 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. |