OF BALANCING THE WORK. 21. The use of the traverse table being explained, we can proceed to compute the area of the ground. The field notes having been completed, rule a new table, below, with four additional columns, two for latitude, and two for departure. Then find, from the traverse table, the latitude and departure of each course, and enter them in the proper columns opposite the station. Then add up the column of northings, and also the col unin of southings: the two sums should be equal to each other. If they are not, subtract the less from the greater; the remainder is called the error in latitude. This error takes the name of that column which is the less. For example, if the sum of the northings is less than the sum of the southings, the error is called, error in northing: but if the sum of the southings is less than the sum of the northings, the error is called, error in southing. We find the error for each particular course by the following proportion. As the sum of the courses Is to the error of latitude, The error thus found may be entered in a separate column; after which add it to the latitude of the course when the error and latitude are of the same name, but subtract it when they are of different names. This will make the sum of the northings equal to the sum of the southings, and is called balancing the work. The northings and south ings thus corrected are entered in columns on the right, under the head balanced. The eastings and westings are balanced in the same manner; the difference between their sums being called error in departure. For an example, we will resume the one already con As 37.25 0.06 10 : 0.02* error in dep. of 1st course. 9.25 0.01 error in dep. of 2d course. 7.60 0.01 0.06: 10.40 : 0.02 As 37.25 0.06 :: : As 37.25 error in dep. of 3d course. error in dep. of 4th course. 22. REMARK I. In finding the error in latitude or departure, for a particular course, the last figure is sometimes doubtful in which case it is best to mark it, as in the third proportion for error in latitude, and the first for error in departure; and then, if the figures taken do not balance the work, let each be increased or diminished by 1. 23. REMARK II. It has already been observed (Art. 18), that if the measurements on the field be correctly made, the sums of the northings and southings will be equal to each other, as also those of the eastings and westings. It is the opinion of some surveyors, that when the error in latitude or departure exceeds one link for every five chains of the courses, the field notes ought not to be relied on. This, perhaps, is a higher degree of accuracy than can be attained. The error, however, should always be made considerably less than one link to a chain. 24. The following is an example in which the latitude and departure of each course have been computed from Instead of balancing by the method just explained, we divide each error by two. Now if we subtract half the error in southing from the column of northings, and at the same time add it to the column of southings, these two columns will exactly balance. In like manner, if we subtract half the error in easting from the column of westings, and at the same time add it to the column of eastings, these columns will also balance. The errors should be distributed in proportion to the lengths of the courses, but this may be done with sufficient accuracy without making the proportions. If any of the courses have been run over rough ground, the probability is that the errors belong to these courses, and they should be distributed among them. In this example we separate the half error in southing into the three parts .40700, .21302, and .04674, and subtract them respectively from the northings of courses 2, 1, and 3, and then place the northings in the balanced columns. For the southings we separate the half error into the four parts .40772, .20031, .03121, and .02752, and add them respectively to the southings of the courses 4, 5, 8, and 7. We then enter the southings in the balanced columns. As the error in easting is so small, we add half of it to the easting of course 3, and subtract half from the westing of OF THE DOUBLE MERIDIAN DISTANCES OF THE COURSES. 25. After the work has been balanced, the next thing to be done is to calculate the double meridian distance of each course. For this purpose, a meridian line is assumed, lying either wholly without the land, or passing through any point within it. It is, however, most convenient to take 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. 10), that all departures in the direction east, are considered as plus, and all departures in the direction west as minus. 26. To deduce a rule for finding the double meridian distances of the courses. Let BC N represent any course, and AB the preceding course; also, let D and E be their middle points. Draw EII, CM, and DG, 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 2EH=2KG, is the double meridian distance of the course AB. H E B G K M C Now, 2DG 2GK+2KL +2LD; but 2KL = IL is the departure of the course AB, and 2LD=MC is the depar ture of the course BC; consequently, 2GD=2GK+ IL+ MC; hence, the double meridian distance of a course, is equal 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. 27. REMARK. 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 dif ference, with the sign of the greater prefixed to it. If the assumed meridian cuts the enclosure, the doublemeridian distances, estimated to the left, must be taken with the minus sign. The double meridian distance of the last course should be equal to the departure of that course. A verification of the work is, therefore, obtained by comparing this double meridian distance with the departure of the course. 28. To apply the above rule to the particular example already considered (Art. 21), rule a new table as below, in which are entered the balanced northings and southings, and the balanced eastings and westings. In this table there is but a single column for the dif ferences of latitude, and a single column for the departures.. The sign shows when the difference of latitude is north, and the Isign when it is south. The sign also shows. when the departure is east, and the sign when it is west 8 |