C—la

69

To compute the mean error the centring error is obtained by using equation (10), and combining the result with tiie mean sighting and reading error of 15" for each angle. A total error of 54" is found for the bearing of the last line. Then using equation (13) for the total mean error at the end of the traverse, the square of the error due to linear measurement is found to be 1289, and the square of the mean error due to angular measurement 5-519. The total mean error is s (1-289 + 5-519) = 2-6—about 1 in 10,000, or about 0-8 per mile. The lines in the above traverse being long is favourable to a high degree of accuracy, and the actual error is about -7 of the mean error, a result that is satisfactory. This example shows that for open country, with traverse lines from about 5 chains long and upwards, the error of the bearing should not exceed I', and the actual closing error in latitude and departure should be not greater than 1 link per mile for each : that is, the actual error should be less than D 414 per mile for the length of traverse. The limits of error permissible in traverses deduced from the last example are not applicable to traverses through broken forest country or for surveys in mountainous localities, where the lines are usually short. In a traverse with forty or fifty lines to the mile the error of reading the chain at each station must be considered, and the sighting and reading error should be increased from 15" to 30" for each line. The reading end of the baud is divided into links, and the reading taken to the nearest tenth of a link, either by scale or by estimation. When a scale divided into tenths is used the greatest error that can occur is o's link, and since the error may have any value between -05 and zero the mean error of reading the chain for each station is therefore '025 link. Taking fifty stations to the mile as an average, the moan error of reading can he found as if it depended on the distance, and consequently combined with the band coefficient. Thus if r denote the roadhig-orror when fifty lines are taken to the mile, and the average line 160 links in length, then — r = ± -025 V 50/8000 = ± T77/ \/8000='002. Combining this with the value of c = '002 gives a coefficient of '0029. Using this value, '0029, for the band coefficient and 30" for the sighting and reading error, with -05 as the greatest displacement in the centring, the mean error can bo computed and serve as a guide to the degree of accuracy attainable in this class of work. Taking the last example, with the lines one-tenth as long as used there, the moan errors are found to hi; as follows : The closing error in bearing is 2' 35" ; the moan closing error is 09. in the case of the angular errors, the error due to imperfect centring is nearly equal to the reading and sighting error. The closing error is 2-71 per mile. for work of this class the error in the bearing should not exceed 3', and the closing error in latitude and departure should not exceed 2 links to the mile, thus giving an actual closing error of 2-83 per mile on the total length of the traverse. The next case to consider is surveys in cities, whore greater accuracy is desired. Such work usually consists in measuring short distances from standard marks, and turning angles off' lines the bearings of which have been accurately measured, When the surveyor, with his staff and instruments, is on the ground the extra time required to measure the distances twice and to take mean bearings is not great. For this work the instrumental constants will be taken as follows : Band coefficient, '0015 — c, ; sighting and reading error, 10" = v ; greatest centring displacement, 0-015 link =a. This band coefficient gives an error of '033 for a 500--band length on the level. The following example is an actual survey on hilly ground, the grades ranging from 30" to 15° 30", the greater number of lines having vortical angles between 6° and B°. In this case all the operations connected with the computation are shown in full.

Example of City Traverse.

The actual closing error in this traverse is 0-01 in latitude and o'os in departure, or a total error of '051 for the traverse of 2956 links. To compute the mean error due to the constants adapted for the theodolite and band, the first step is to determine the centring error from equation (10).

a o d c2 Observed Bearing, Measured Distance. Latitude. Departure. Total Latitude from No. 1. Total Departure from No. I. I links. 1 2 3 4 j 5 6 7 8 1 165 lJ 46' 24" 130 30 00 L85 30 00 232 00 00 320 40 00 271 31 00 329 34 15 59 34 .15 390-99 535 00 250-00 140-00 10000 228-69 882-20 429-25 000 - 379-00 347-45 248-85 - 86-19 + 77-35 + 6-05 ■+- 760-68 + 217-40 o-oo + 9609 + 406-82 - 23-96 - 110-32 63-38 - 228-6.1 - 446-8.1 4- 370-12 0-00 - 379-00 726-45 975-30 - 1061-49 - 984-14 987-09 - 217-41 001 o-oo + 96-09 + 50291 ■f 478-95 + 368-63 + 305-25 + 7(474 - 370-17 0 05