ing in the same latitude the whole way, it becomes necessary to run it as a curve. (See Sec. 209.) Now the amount of this increase in bearing from the north is equal to the convergence of the meridians between the two places, so that in the case of A and B the difference in the bearings of the same straight line obtained by observation at each place will be represented by the angle BPA, which for ordinary work we may consider equal to the difference of longitude of the two places multiplied by the sine of their mean latitude. (See note F, Appendix.) Thus if in latitude 40° north we start a straight line from A due west and run it to C through 1o of longitude, the bearing obtained by observation at C should be S. 89°, 21′ W. But since it often needs some calculation to ascertain the difference of longitude, we can best proceed in ordinary work by finding from the following table the correction to be applied. Thus if in latitude 50° N. we have run a line which gives a total amount of easting or westing (i.e., Total Departure) equal to 60 miles, the amount of the correction to apply will be TABLE OF CORRECTION FOR CONVERGENCE FOR 1 MILE OF EASTING OR WESTING. This shows the necessity, when running a long continuous survey, of referring all bearings to an Initial Meridian, either at the point from which the survey started, or at a point near its centre. The same remarks of course apply to magnetic courses to a certain extent, but in this latter case, on account of the constantly changing variation, such corrections are hardly practical. 59. When the transit-line crosses a river or ravine or some other obstruction over which it is difficult to obtain direct measurement, the best way to proceed is by Triangulating, using whichever of the methods shown in Fig. 13 is most applicable to the case. A FIG. 13. The angles at A and Feach = 90°, and at J, K, and L = 60°; then If the ground on which we measure our base has a tolerably uniform slope in the direction of the base, it is better to take direct measurement along the surface of the ground and multiply the distance so obtained by the cosine of the inclination to obtain the horizontal distance, than to "break-chain." Whatever difference in elevation there may be between two such points as A and B, if the base measurement is reduced to the horizontal, the distance as calculated for AB, from the angles observed with a transit, will also be the horizontal dis tance. If the angles were observed with a sextant, of course this would not be the case. (See Sec. 144.) If, instead of encountering such obstructions as those given above, an obstacle which we are unable to see across presents itself, such as a huge detached rock on which we cannot set up the instrument, then perhaps as good a way as any to get round it is by offsetting the line so as to run past it on a parallel one, and then on the far side, by equal offsets, getting back on to the former line. If the obstacle, however, is too large to pass it well by this means, we can apply the equilateral triangle JKL (Fig. 13). This latter method is a good one to use whenever practicable: there is no calculation necessary in connection with it, the angles used are those most favorable to exact work, and where the obstacle can be seen over, a check can be applied by observing the angle at K. After having run the line a certain distance ahead, represented by the amount L, it is often necessary to "back-up" and start the line again from the instrument so as to strike a point a certain distance d on one side of the point where the first line struck; the correction C for this may be found thus: For more on the subject of triangulation, etc., see Part III. 60. The LEVELLER'S WORK on preliminary location consists mainly in taking the elevation at every full station, and at any intermediate points where he may consider it advisable to do so. The best form of keeping notes on such work is the following: H.I. in any line = Elev. B.S. in preceding line, The "Intermediate" column is sometimes omitted, but the insertion of it makes it easier to check each page by means of the difference of the sum of the Back-sights and Fore-sights. To apply this check between two stations, A and B for instance, which have been used as turning-points, add together all the back-sights between A and B (including the B.S. at A, but excluding it at B); then add together all the foresights (excluding the F.S. at A, but including it at B): the difference of these two sums should equal the difference in elevation of A and B. If the sum of the back-sights is greater than the sum of the fore-sights, B is higher than A; but if less, then lower. The levels should be worked out in the field whenever time permits, for reference on the work. The profile for each day's work should be made out when possible in the evening of the day on which the work was done. As regards the precision of a line of levels run as above, the probable error is usually assumed to vary as the square root of the distance. The limit on the British Ordnance Survey is 0.01 foot per mile; the U. S. Coast Survey requires a limit of 0.03 per mile. If we assume a limit of 0.05 per mile for rough work, the probable error for any distance equals 0.05 mile. Thus in 100 miles the probable error = 0.50 ft. the subject of levelling see Parts II and III. For more on 61. The TOPOGRAPHER'S WORK consists principally in taking the ground slopes, with more or less accuracy, at every full station and at any intermediate points where he may consider it necessary, by means of which a contour plan may be constructed. To do this he obtains from the leveller the elevation of each station and plus station at which he has taken levels. There is a variety of methods in use of obtaining the slopes, and the advantage of each depends on the accuracy required, the nature of the country, and the vertical distance apart of the contour-lines. Where the slopes are steep and accurate work is wanted, a 10-foot slope-rod with clinometer gives very good results, but is a cumbersome sort of instrument to carry about. Where 5-foot contours are wanted, a hand-level is very con venient, since by considering the height of the eye above the ground to be 5 feet, the point corresponding to each contourline is located at once by the level,-5 feet being an easy height to which to accommodate one's self,-and by pacing the distance between these points we have thus simply to enter the distances in the notes through which each contour passes. By taking the alternate points selected in this way, this method is of course equally applicable to 10-foot contours. shows how this method is worked. Fig. 14 Elev. FIG. 14. Suppose, e.g., that for a certain station the topographer obtains from the leveller the elevation of 1823.8, and that he is taking 5-foot contours. Then, if the ground is as shown in Fig. 14, he proceeds as follows: The contour-line nearest to this elevation is that of 1825 feet, the plane of which passes about 1 ft. above the ground-level at the station, so that by standing at the point a he can estimate with his eye the amount of 1.2 feet, and thus find the point b which corresponds with the contour of 1825. Similarly, standing at b he finds c, and so on up the slope as far as he considers necessary. Then returning to a, he works in the same way on the lower side. If the distances are wanted accurately, he should have a man with a tape to assist; but as a rule, pacing, where it is practicable, gives good enough results. The only notes to be kept in this case are the distances out (right or left) to the respective contours. An Abney hand-level (with vertical arc) is also frequently used, and gives good results. All methods, however, which involve taking the angles of the slopes themselves necessitate extra work. One method of reducing this amount of labor is to have a set of scales for the various slopes, each made proportional to the cotangent of the inclination; but by the use |
