Page 2 Advertisements Column 4

Wellington Independent, Volume XXIV, Issue 2806, 6 March 1869, Page 2

levelling of the instrument has been attended to, and the observations carefully taken. (See appendix for form of field book.) Vertical Angles. 26. Vertical angles are to be observed to all principal stations upon both faces of the instrument; the mean readings will furnish the true angles of elevation or depression. The height of the instrument above the ground at the time of observing, and also of that part of the object observed to, should be recorded in the field book. Base of Verification. 27. Similar observations are to be taken at every station until the base of verification is arrived at. This second base line must be measured with the same amount of care as was bestowed upon the first. The triangles comprised between these two bases will then form a complete series, and the computation of the sides should be forthwith commenced. Geodetical Angles. 28. The differences at any station between the mean bearings of any two other stations observed therefrom will be the Spherical or more properly the Spheroidal angle, included by the planes of the great circles joining the station observed from and the stations observed to. Since all three angles of every triangle used for computation must necessarily be observed, it follows that if correctly measured the sum of the angles should be 180 ° + e, where e denotes the Spherical excess which may be easily computed from the formula e=a² sin B sin C/2r² sin A sin 1 a being the side comprehended between the angles B and C, and r the mean radius of the earth; constant log. 1/2r² sin 1 = 0.3735695 adopting Professor Airy's value of the mean radius of the earth. Legendre's theorem for the computation of small geodetical triangles. 29. Legendre's Theorem proves that if one third of the Spherical excess be deducted from each angle of a triangle, and then computing with the angles so diminished with an arc base, the results give arc sides equally as accurate as when computed by the rigid and very laborious process of using chord sides and angles. This theorem holds good with triangles of 450 mile sides. The Spherical excess amounts only to 1" in 70 square miles; now as the largest triangle of the Major Series does not exceed this area, it may safely be omitted in our operations. Errors of observation in angles to be expunged. 30. In practice with an eight inch theodolite the sum of the angles of a triangle generally vary on an average some 10" from 180° + E, or say from 180° omitting the Spherical excess. This difference exhibits the amount of error made in the angular measurements, and may arise from a number of unavoidable causes. It is obvious that to obtain satisfactory results for the succeeding computations the geometrical conditions of the triangles must be satisfied by a system of correction dispersing these errors. The easiest method is to apportion one third of the triangular error to each angle, thus making their sum equal to 180°; but the best, and the one to be recommended is that explained in Par. 8, Page 322, Galbraith's Trigonometrical Surveying and Levelling. System to be adopted in the order of computation. 31. Unless a system is also adopted for the computations the results will be found to vary, perhaps even so much as prima facie to destroy confidence in the correctness of the work. It must always be borne in mind that accumulation of error is less liable when the computations are as few and direct as can be performed, therefore operose, circuitous, and complex modes, intelligible only to the computer, should be studiously avoided. Detection of error on closing sides, 32. Triangles arranging themselves in a polygonal figure, as BCDEFG, having a common apex at A, afford an easy and effective check upon the accuracy of the work. If AB be the given base it is obvious that by successive calculations of the triangles round the circle the side AB should return exactly to its measure, provided the triangular errors before mentioned have been so dispersed as to satisfy all equations of condition. But as this will rarely be the case the difference exhibited between the side AB, and its calculated measure on return, if within certain limits, may be eliminated on the principle of a gradual accumulation of error. It is quite practicable that this limit should be within the proportion of one link to every 100 chains of the length of side. Elimination of Errors. 33. In order that the angles and sides should agree after eliminating the errors in each proceed in the following manner. The sum of the angles round the apex at A must be made equal to 360°, and the sum of the three angles of each triangle to 180°. Let AB be the base from which the computations emanate, then the sides AC, AD, AE, &c, in turn become new bases for continuation. Compute these round the circle and compare the difference between the Log.AB and that obtained by computation from the last base AG, call the difference d. Generally Log. AC=Log.AB+Log. sin B + Log. cosec C, but since Log. AC has to become