adjusted traverse lines will not differ very materially from the lengths actually measured on the ground. For instance, as cc does not differ much from do, the length CD will not differ materially from cd. Similarly as do is nearly equal to eE, DE will be nearly equal to de. Adjustment of Closing Error by Calculation. When the latitudes and departures of a closed traverse are calculated, the algebraic sum of the latitudes should be zero and the algebraic sum of the departures should also be zero. This is evident, as an inspection of Figs. 81 or 82 will show that the traverse goes first a certain distance north to the point K and then returns an equal amount south to the starting point a. The sum of the northings is therefore equal to the sum of the southings, i.e., the algebraic sum of the latitudes is zero. Similarly with the departures. For the reasons already stated the algebraic sums of the latitudes and of the departures will seldom equal zero and have to be adjusted. Example of Adjustment of Closing Error by Calculation.—The following example of a closed traverse, Tables I. and II., will show the method of adjustment when the latitudes and departures are calculated. The calculations of the latitudes and departures were made as already described on pages 93 and 94. CLOSED TRAVERSE-TABLE I. The lengths and bearings being as given in Table I., upon calculating the latitudes and departures it is seen that the positive latitudes are 9.2 in excess, while the negative departures are 14.8 in excess. This is called the "closing error," and requires to be adjusted. To do this, apportion the error proportionally to the lengths of the sides. For instance, the length of the side AB is 1,060 ft., and the total length of the sides being 5,729 ft., we have— 1060 Correction of latitude of AB = × 9.2 = 1.7 ft. 5729 As the latitude of AB is positive, and the positive latitudes are in excess, this is to be deducted from the latitude of AB, and we get— Corrected latitude of AB = 1015.5 − 1.7 = 1013.8. Similarly the correction for BC is Line. 5729 value of the latitude of BC is 300.3 - 1.9 298.4. The correction ILIO for CD is × 9.2 = 1.8, and as the positive latitudes are in excess, 5729 this falls to be added to the latitude of CD, and the corrected value is 390.2+1.8392. Similarly the corrections for the other latitudes are calculated and applied in the same manner, the departures are treated in exactly the same way, and we get the corrected latitudes and departures as given in Table II. CLOSED TRAVERSE (CORRECTED)—TABLE II. AB BC CD DE EF FG Bearing. I 202 343° 20' 75° 32' 110° 35' 221° 3′ 280° 20' 233° 30' Totals Length, × 9.2 = 1.9, and the corrected 1,060 1,202 1,110 850 802 705 5,729 Upon adding the latitudes and departures we see that the sum of the latitudes is 0.1 and the sum of the departures is o. The error of o.I still remaining in the latitudes is owing to the corrections being calculated to one place of decimals only. The traverse is now to be plotted by ruling a meridian through the starting point a and proceeding in the same manner as described for the plotting of the traverse on Plate III., Figs. 79 and 80. The correction of the latitudes and departures evidently affects the bearings and lengths of the lines AB, BC, &c. These may be measured with the protractor and scaled after the work has been plotted from the corrected latitudes and departures, or, if more = accuracy is desired, the new lengths and bearings may be calculated from the corrected latitudes and departures. Thus the new length of any of the lines is obviously and the tangent of the angle made with the meridian corrected latitude = Adjustment of Closing Error when some of the Measurements may be considered more accurate than others.-The above method of correcting and adjusting the closing error is based on the assumption that the chaining and the measurement of the bearings are equally in error, also that the relative accuracy of the lengths and bearings of all the lines is the same. When from special circumstances the measured lengths and bearings of some of the lines may be considered to be more accurate than others, the following method may be adopted. Take one of the lines as a standard and assume that the error in this line is 1; from this basis estimate what the probable error in each of the other lines would be, taking into consideration the special circumstances of each, such as any particular obstacles to measuring, roughness or steepness of ground, number of observations made to determine bearing, if check measurement of length made, and so on; these probable errors are to be for a distance equal to the length of the standard line. Each of the lines being thus weighted with its probable error as 1.5, 2, 3, 5, &c., multiply the length of each line by its probable error and then we have Correction of lat. or dep. of any line = multiplied length of given line sum of all multiplied lengths x whole error in lat. or dep. Adjustment of Closing Error when the Error is considered to be due to the Chaining only.-When it is considered that the bearings are practically correct, and that the closing error is due to the chaining alone, the correction of each line is to be computed as follows :— Correction of lat. or dep. of any line = _given lat. or dep. sum of lats. or deps. × whole error in lat. or dep. The closing error may be assumed to be entirely due to the chaining when the bearings have been carefully measured with the theodolite and the bearing of the first line, when redetermined at the close of the traverse, is found to agree very nearly with its true or assumed bearing at the beginning of the traverse. Amount of Closing Error allowable in practice.*-For purposes of comparison the closing error is taken as the ratio of the length of the line joining the initial and final points of the traverse (as plotted or computed from the field notes) to the length of the whole perimeter of the traverse. In ordinary open country the closing error should not exceed 1 in 300. In town work the closing error should average about 1 in 5,000. For special purposes where greater accuracy is required, as in tunnel work, &c., the precautions in measuring the lengths and angles described in Chapters VII, and XII. must be adopted. Compass Traverse Surveys.-A traverse survey executed with the compass only, e.g., with a circumferentor or surveying compass (Fig. 60), or with Whitelaw's theodolite (Fig. 61), is executed in precisely the same manner as those already described. The essential difference is that the bearing of each line is measured from the magnetic meridian, and angular errors are therefore not cumulative, as they are in the preceding methods. A long traverse hastily executed with the circumferentor will therefore be in general more accurate than the same traverse if hastily executed with the theodolite, and indeed the speed and accuracy attainable in compass surveys is remarkable compared with theodolite surveys. At each set up of the compass the bearings of two lines may be taken, the line in front and the line behind, and when great speed is necessary it is usual to set up the compass over every second station only. To guard against local attraction, however, the compass should be set up at each station, and the back bearing as well as the forward bearing of each line observed. If the back bearing does not differ by 180° from the forward bearing there has either been a mistake in reading the forward bearing or else there is some local attraction. In the latter case the angles between the lines must be booked in place of the bearings, until the correspondence of the back and forward bearings indicates that the attraction has ceased. The |