a—ia

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tn selecting the mean error as a test of the accuracy of field measurements, the same test must be applied to determine all the instrumental constants. Thus the mean error of reading and sighting of a theodolite and the mean error in the chaining constant must be used throughout. If either the probable error or the error of mean square is used a different set of instrumental constants will ensue, but all leading to the same final result. From the table of constants the relation is,— B m - I'lß3 E v (2) E m = 0-798 E. (3) It is necessary to remark that the above errors are true errors, as distinct from apparent errors. For instance, the difference between the sum of throe angles of a triangle and 180° is a true error, but the difference between each value of a series of measures of a baso-line and the arithmetic mean is an apparent error, because the true length of the line is an unknown quantity. The values are, — E m Apparent mean error of arithmetic mean = + !" (.1) Vu Ej m True mean error of arithmetic mean = + , - (5) Vn — 1 In fixing the limits of error permissible in field-work the accuracy aimed at should not be difficult to obtain with the steel tape and theodolite, without special apparatus. A spring balance to register the pull and a thermometer to give the the temperature are necessary. The errors in traversing are due to two causes : (I) Errors in the linear measurements; (2) errors in the angular measurements. Cumulative error in linear measurement is represented by— c = c v ' I (6) where c is a constant depending on the apparatus used, and I the length of the line. For a steel band, on the level", using plummets to effect the marking of the terminals, the value of c has been determined to lie between + 0'0()15 and ± o'oo2o. Adopting the value 0002, and taking a stretch of 5 chains, gives, from (6) — c = + -022 VSOO = + '045 ■ (7) The above result is open to criticism, but most chaiumen would undertake to measure a distance and keep the marking of the separate lengths within the above limit without any special precaution. The result in (7) includes the error due to inaccurate tension, error due to imperfect alignment, and the personal errors of the chainmen, &c. When the measured distance is inclined to the horizon the effect of the errors due to angular reading and refraction have to be determined and combined with the result in (7). For lengths between 5 and 10 chains the value of refraction can be taken as 30". Angles of elevation will be 30" too great, and angles of depression 30" too small. In taking these observations in the field a forward reading is usually taken for the first band-length and a backward sight for the second length. If, then, the grade does not change sign (that is, change from, say, uphill to downhill) the effects of refraction will balance each other. In cases where vertical angles are large, and close readings are required, reciprocal angles can be taken without any undue expenditure of time : thus the refraction can be eliminated by suitable methods of observation, and will not be included in this investigation, The error due to imperfect reading and sighting of the vertical circle of a 5" theodolite in good adjustment, for the purpose of obtaining the slopes, can be taken as I. The average slope of all the lines of a traverse to embrace all kinds of surveys is difficult to determine. For road traverse the grades are confined by regulation, and an average of 3° can be taken. The result of experience is that the number ol lines with slopes under 10° is very much greater than the number of lines with slopes over 10°. By using an average slope of 10° for the total length of the traverse a larger mean error will be found than if the correct average were used, except in the case of surveys in stsep and very broken country. The reduction formula is— If = I cos 0 (8) Where / is the included distance and 0 the inclination, the mean error being— -E H = ± ✓ (Ef cos a 0 + E,V sin 8 0) (9) taking 6 = 10°, I = 500, El = ± (105, E c = I', E B = ± -05 (9) that is, the mean error of measuring a single baud length for any line, level or inclined, is + '05. The errors in the horizontal angles come under the headings— (a) Errors due to sighting ; (b) errors due to reading the vernier ; (c) errors due to imperfect centring. The error due to sighting depends very much on atmospheric conditions, and to a lesser extent on the length of the lines, and a constant mean value for all lines can be used. The reading-error is independent of the length of the lines, and the errors of sighting and reading can be combined and a constant average error of 15" will be used throughout. The mean error in reading any theodolite can be found by an examination for the errors due to eccentricity