C—la

68

and the errors in graduation of the divided circles, by using the constant angle between the two verniers ami finding the readings at intervals of 5° or .10° from 0° to 360°. From these readings the exact angle between the verniers can be determined and the eccentricity can be computed. Then by correcting the readings for eccentricity the remainders represent the errors of graduation and reading. Surveyors often find that an instrument used for measuring the angles of a traverse, by reading one vernier only, consistently gives the bearings of the lines too great or too small when compared with a check-bearing, and no amount of repeating the work will disclose any appreciable error. Such results are caused by eccentricity, or the centres of 'the axis of rotation and the divided circles not being coincident. This error is eliminated by taking the mean of both verniers, or by finding the error due to eccentricity, which may amount to 10" in some cases, and correcting each angle. c The error due to imperfect centring is almost negligible for long lines, but it N. increases very rapidly as the lines shorten. If a maximum error ris decided on. and A,B,C, three consecutive stations of a traverse, then the mean error of centring is— N u =± 3. 4 AI. A EC . - 10 ) / A (For the mathematical investigation of the error of centring, see " Effects of Errors in Surveying, by IT. Briggs). To determine an average value for r, consideration has to be given to the plummet, and deflections by wind or other causes. A value of r = '05 will bo used. If the angle ABC is denoted by a, AC 2 = v' (AB 2 + BC 2 - 2 AB ■BC cos a) This is greatest when a = 180", and therefore the mean error of ranging a straight line is greater than that of making a traverse with the same number of stations or sights, a result proved by experience. Having decided on the moan value of the sighting and reading error of the instrument as 15", the maximum centring displacement as '05, and the coefficient for the band c = '0022, the mean error of any traverse can be computed and compared with the actual closing error of the survey. If the actual closing error is not greater than the computed moan error the work can be considered as satisfactory. If, however, the closing error is greater than the computed mean error, a revision of the survey should be made. The following is from a closed survey by steel band and 5 in. theodolite over hilly country. Denoting the length of the lines by /,, 12,l 2 , /,,, &0., /„, the bearings of the lines by Bj, 83,B 3 , 88,B B , &c, B,„ the mean error of idle bearings by /;,, b. it b., &c, />,,, the mean error in the latitude of the end point— = ± V j'C 8 (/, cos- Bj -1 /;■' cos- 8., -f / n cos- B„) + (/,,- 1,.; sin- 15, -1- I* tV sin- B, + l\ b\ cos 2 B B )j (11) The mean error in the departure of the end point - = + \/-]C- (/, sin a B, + I, sin" Bj + l\ sin ß B n ) + l\ b\ cos 2 .15, +l\ b\ cos- B a + l\ b\ cos 2 B n ) f (12) The mean error at the end of the traverse— =± -/ { C 2 (I, +I, + /„) +l\b\+l\b\+ l\ b\ + P„ b\) \ (.13) Example of traverse over hilly country

„ e ? Bearing. Distance. Number. b Links. . 1 271° 49' 20" 1117-2 2 ! 278 31 00 10930 3 i 346 25 00 3916 4 I 314 37 00 2017-8 5 i 280 31 00 514-4 6 235 27 00 898-7 7 203 08 00 483-2 8 227 57 00 869-6 9 340 25 00 422-6 10 8 33 20 3039-5 11 19 14 00 3913-7 12 90 00 00 4209-2 18 ■ 180 00 00 7652-3 Latitude. Departure. Total Latitude. I Total Departure, ! I I || + 35-5 - 1116-6 + 35.5 1116-6 161-9 1081-0 197-4 2197-6 380-6 92-0 5780 2289-6 14173' 1486-3 1995-3 3725-9 + 93-9 505-8 2089-2 4231-7 509-7 74.0-2 1579-5 4971-9 444-3 189-8 L185-2 5161-7 582-4 645-7 5528 5807-1 + 398-2 - 141-6 951-0 5949-0 3005-7 |- 452-2 ;. 3956-7 5496-8 + 3695-2 1289-3 7651-9 - 4207-5 + 4209-2 + 7651-9 : + 1-7 - 7652-3 ... 0.4 +C7 Total ... 26622-8 ■ The actual error in latitude is v { (0-4) ! + (1-7)' } =- L75. 0 4, and in departure 1'7, therefore the total actual error is