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This error for latitude or departure must be distributed among the latitudes or departures of all the courses, in proportion to the length of each course, observing to add the correction, when applied to the deficient column, and to subtract it, when applied to the other.
We will illustrate this, by the example of (Art. 75).
The error in southings, 3 links, is to be distributed among the northings and southings, in proportion to the lengths of the courses; a part to be added to the southings, and the remaining part subtracted from the northings. The error in westings is similarly distributed among the eastings and westings. For this, two new columns are formed, called, the balanced latitudes and departures; and to these columns the latitudes and departures are transferred, after the corrections have been made: the north latitudes being marked +, and the south latitudes, in order to distinguish them readily, and also, for convenience in the calculations which follow.
The error of .03 in the latitudes is distributed among the latitudes, by subtracting 1 link from each of the northings of courses 1 and 2, and adding 1 link to the southing of course 4. This produces a balance.
Of the error of 4 links in the departures, 1 link is added to each of the departures west, and 1 link subtracted from each of the departures east. This produces a balance.
NOTE. When a knowledge of the conditions under which the survey was made, enables us to determine that errors were more likely to occur at certain points, it is doubtless best to apply the corrections to those courses where it seems probable the errors were made.
88. The limit of error, to be allowed, depends of course upon the importance of the survey.
In ordinary farming districts, the error should be as small as 1 link to 5 or 10 chains of perimeter.
The "error of the survey" should be considered as the length of the line necessary to close the boundary, and is equal to the square root of the sum of the squares of the errors of latitude and departure. Thus, in the above example, the error of the survey is 5 links. The perimeter being 37.20 chains, the error is about 1 link to 7.45 chains, or of the perimeter.
89. It will be well to bear in mind the fact, that if the error in the perimeter has been made in one course only, and distributed by the ordinary methods of balancing, among all the courses, the error in area will be larger than the error in perimeter.
90. When the error is so large that a re-survey becomes necessary, the balancing should be carefully re-examined.
In many cases, the location of the error may be determined by inspection of the computation, and a portion of the labor of a re-survey, thereby saved.
This refers more particularly to those cases where the error is one of chaining, and is mostly in one course. Errors of this
the chain properly between the courses, but make occasionally an error in counting the fractional part of a chain at the end of a course.
In such cases, the location of the error may be detected by observing, first what columns contain errors, and secondly the ratio of the errors of Latitude and Departure.
When the error in the survey has been a single one, of distance only, then the ratio between the errors of Latitude and Departure must be the same as the ratio between the Latitude and Departure of the course to be corrected. If the errors be in northings and westings, then the courses running either North and West, or South and East, should be examined.
91. The surveyor should take every possible precaution against errors in the bearings. This is accomplished by backsighting, taking bearings of some one object from several stations, and also by taking bearings of stations across the field. These precautions will give, in general, sufficient data for the detection of an error in bearing; for, by mapping the survey, and drawing the lines to indicate the extra bearings, the error is revealed by the failure of the lines to meet at a common point.
92. One source of error, in large surveys with the compass, is frequently overlooked. This is the diurnal variation there is sometimes as much as 15 minutes variation during the daylight hours.
Errors from this source can only be avoided by testing the compass, at intervals of two or three hours, by taking the bearing of the same line.
93. If each of the angles of the survey, included between two consecutive courses, be calculated by the method explained in Article 102, the bearings may then be verified by comparing
the sum of these angles with the sum of the interior angles of any polygon (Leg., Bk. I., Prop. 26). The same verification may also be made when the angles are measured with the theodolite or transit.
94. There is one kind of error frequently made in reading the compass when the bearing is nearly east or west. The error arises from reading North for South, or the reverse. If the survey is otherwise correct, the error in latitude is just twice the latitude of the course containing the error.
DOUBLE MERIDIAN DISTANCES.
95. After the work has been balanced, the next thing to be done is to calculate the double meridian distance of each
For this purpose, any meridian line may be assumed. It is, however, most convenient to assume that meridian which passes through the most easterly or westerly station of the survey; and these two stations are readily determined by inspecting the field-notes.
Having chosen the meridian, let the station through which it passes be called the principal station, and the course which begins at this point, the first course. Care, however, must be taken, not to confound this with the course which begins at station 1, and which is the first course that is entered in the field-notes.
It has already been remarked (Art. 71), that all departures in the direction east are considered as plus, and all departures in the direction west as minus.
96. To deduce a rule for finding the double meridian distance of any course. Let SN be the assumed meridian. Let BC represent any course, and AB the preceding course; also,
perpendicular to the assumed meridian NS. Draw also AI, EK, and BL, parallel to NS. Then 2DG is the double meridian distance of the course BC, and 2EH2KG, is the double meridian distance of the course AB.
Now, 2DG 2GK + 2KL + 2LD; but
2KL IL is the departure of the course AB, and 2LD = MC is_the_departure of the course BC;
2GD = 2GK + IL + MC;
hence, the double meridian distance of a course is equal to the double meridian distance of the preceding course, plus the departure of that course, plus the departure of the course itself: if there is no preceding course, the first two terms become zero. We therefore have the following
RULE.-I. The double meridian distance of the first course is equal to its departure:
II. The double meridian distance of the second course is equal to the double meridian distance of the first course, plus its departure, plus the departure of the second course:
III. The double meridian distance of any course is equal to the double meridian distance of the preceding course, plus its departure, plus the departure of the course itself.
NOTE. It should be recollected that plus is here used in' its algebraic sense, and that, when the double meridian distance of a course, and the departure which is to be added to it, are of different names, that is, one east and the other west, they will have contrary algebraic signs; hence, their algebraic sum will be expressed by their numerical difference, with the sign of the greater prefixed.