first course and the angle turned are to the right, the sum of the bearing and angle should be the bearing of the second. course. The bearing of a course, as thus computed from the bearing of the adjacent course and the angle between the courses, is called its deduced or calculated bearing. The deduced bearing of the course BC is, therefore, N (68° 15′ +10° 25′) E, or N 78° 40′ E. The magnetic bearing of the course BC, as read from the needle, is N 78° 30' E, which agrees with the deduced bearing within the limits of accuracy of the compass.

43. Field Notes.-A good form for the field notes of a deflection traverse like that described in Art. 41 is given below. The first four columns are supposed to cover some, or all, of the left-hand page of the notebook.

The last

column represents the entire right-hand page, which is used for remarks and sketches. The sketch shown locates Station 0 and describes the direction of the first course by means of the angle that it makes with the center line of the railroad at this station.

Another way of recording the work is simply to make a sketch on which angles and distances are marked, with whatever explanations are necessary.

TRANSIT SURVEY OF A CLOSED FIELD

THE FIELD WORK

44. Measuring the Angles.-The corners of the field to be surveyed having been marked, the relative directions of the boundary lines are determined as for a traverse, setting the instrument at the different corners in succession, and using either the method of azimuths or that of deflection; the former is by far the better of the two. If the direction of the true meridian is known, azimuths are referred to it. If not, a reference line, defined by permanent marks or monuments, may be assumed as a meridian, to which all azimuths are referred.

Another method consists in setting the instrument at the different corners in succession and measuring directly the angle between the two lines meeting at each corner.

45. Field With Irregular Boundary.-When one or more of the boundary lines are not straight, as in fields bounded by rivers or lakes, a traverse line is run as close to the boundary line as possible, and offsets are taken from the traverse line to those points at which the direction of the boundary line changes.

46. Checking the Angles.-After the work is finished, the accuracy of the measurement of the angles is checked by one of the following methods, according to the method used in measuring the angles:

(a) When the Angles Are Measured Directly.—As explained in Geometry, Part 1, the sum S of the interior angles of a polygon of n sides is given by the formula

S = 180° X (n − 2)

It should be borne in mind, in applying this formula, that reentrant angles, as that at A, Fig. 24, are greater than 180°.

The angle A should be called 260°, not 100°. The sum of the measured angles should satisfy the formula within about 2 minutes per angle.

(b) When the Deflection Method is Used.-Each deflection angle, being the angle made by a side with the prolongation of the preceding, is an exterior angle of the polygon. The sum of all these angles should be equal to 360°, within the limits mentioned above.

(c) When Azimuths Are Used.-The azimuth of the first line is measured, or laid off, when the transit is set at the first corner, and then the azimuth of each line is determined from that of the preceding, as al

ready explained. In this manner,
the azimuth of the first line may be
determined from that of the last, c<
by again setting the transit at the
starting corner. This second de-
termination of the azimuth of the
first line should be approximately
equal to the one with which the
survey was started. Besides, the B
sum of the angles of the polygon

should satisfy the formula already

given. The manner of determining

-100°

092

FIG. 24

the angle between two lines whose azimuths are known is similar to that used when the bearings are known, and does not require special explanation.

BALANCING THE SURVEY

I. WHEN THERE IS A SMALL ANGULAR ERROR

47. Latitude and Longitude Ranges.-The latitude and longitude ranges are computed by the formulas in Art. 39. If the azimuths have not been measured directly and the azimuth of one of the sides is known, the azimuths of the other sides can be readily determined from the measured angles. If no azimuth has been measured, any of the sides

is assumed as a meridian, and azimuths and ranges are referred to it.

48. General Remark on Balancing.-As in the case of compass surveying, it will generally be found that a transit survey does not close. Here, however, the method of balancing explained in Compass Surveying, Part 2, cannot be applied, as that method assumes the possibility of larger errors in the angles than are likely to occur in transit work.

49. Case Where the Angular Error is Small.—When, in testing the angular measurement of a survey that does not close, it is found that the angular error, though sufficiently great to be considered, is small as compared with the error of closure, the following rule may be applied for correcting the ranges:

Rule.

As the arithmetical sum of all the ranges of one kind is to the corresponding range of any course, so is the total error in the sum of the ranges of that kind to the correction to be applied to the corresponding range of that course.

In order to express this by a formula,

Let R =

r =

E

=

с =

arithmetical sum of ranges of one kind;
corresponding range of any course;

total error in latitude or longitude ranges;
correction in latitude or longitude range, to be
applied to r.

Then, R: r = E : c; whence,

If it is desired to weight the courses when using this rule, the ranges of each separate course are multiplied by the weight assigned to that course, and the ratio between each weighted range and the arithmetical sum of all the weighted ranges of the same kind is used in distributing the error. In other words, the weighted range is substituted for and the arithmetical sum of the weighted ranges for R in the above formula.

The student should notice that only arithmetical values are used in applying the formula. The correction for any range

is added arithmetically to the original range, if that will diminish the total error; otherwise, it should be subtracted. Thus, if the sum of the northings is greater than that of the southings, the northings should be diminished, and the southings increased.

II. WHEN THERE IS NO ANGULAR ERROR

50. General Consideration.-The foregoing rule for balancing a survey is founded on the assumption that the error of closure is due partly to erroneous angles and partly to errors in chaining. When, however, the angles are measured accurately, the error of closure may be assumed to be mainly due to erroneous chaining. In this case, the survey must be balanced by correcting the lengths of the sides, which must be so adjusted as to give a closed figure approaching the true conditions as nearly as possible. In adjusting the lengths of the lines due consideration should be given to the following principles:

Principle I.—Measurements made either up or down a slope are likely to be too long as compared with measurements made under similar conditions on level ground.

Principle II.—Error in chaining is more likely to occur in lines measured over rough ground or under unfavorable conditions than in lines measured over smooth ground and under favorable conditions.

These principles may serve as a guide in balancing a transit survey, an operation that must be done by trial, as no exact method has yet been devised.

51. Trial Method.-Let ABCDEF, Fig. 25, represent the plat of a survey that contains no angular error and does not close, and in which the line AF represents the error of closure.

From a mere inspection of the plat it is usually easy to determine which lines must be lengthened and which lines shortened, in order to close the survey. Should any difficulty be experienced in determining this, prolong the closing line,