CHAPTER XVIII. ON THE COMPUTATION OF LATITUDES, LONGITUDES AND AZIMUTHS OF TRIGONOMETRICAL STATIONS. LET A and B be two Trigonometrical Stations. The latitude and longitude of A, together with the distance of ▲ to B at the sea level, and the azimuth of the line as appears to an observer at A, being given, it is required to deduce the latitude and longitude of B, and the azimuth of the same line BA as appears to an observer at B. The symbols which are usually made use of, to represent the elements given, as well as those required, are as follows: Elements given. λ Latitude of A L Longitude of ditto A Azimuth of B from A* C Distance from A to B. Elements sought. X Latitude of B L' Longitude of ditto B Azimuth of A from B. Of the foregoing seven symbols, two only, namely, A and B, which stand for azimuths, require some explanation. In the Revenue Survey the origin of the azimuthal arc is placed in north, whence it proceeds by east to south, and thence again it returns by west to north. This is the common mode of reckoning the azimuth. * The method of deriving the first or fundamental azimuth at the origin of the triangulation, will be found in Part V. In the formula which will be given hereafter, the azimuthal arc will be taken to commence from south, and to proceed by west and north, round the whole circle of the horizon, as observed in the Great Trigonometrical Survey. According to this view, the azimuth of west will be 90°, that of north 180°, and lastly, that of east 270°. It is necessary to mention at this place that there are two solutions of the problem under consideration, the spherical and the spheroidal. In the former the earth is supposed to be a sphere, in the latter it is taken as a spheroid. In this work we will adopt the spheroidal solution, in the first place, because it is more consonant to truth than the other, and secondly, because the process of computation it gives rise to, has been arranged by Col. G. Everest, late Surveyor General of India, into a form which is susceptible of easy and convenient application to survey operations. In the computation of the Great Trigonometrical Survey of India, the dimensions of the earth supposed to be a spheroid are taken at the following values: Axis Major a = 20922931-8 feet. These elements are derived from a comparison of the Dodagontah arc, comprised between Punnæ and Kalianpur, measured prior to the year 1826, with the French arc beginning at Greenwich and ending at Formentera. On a slight consideration it will be evident that if the differences (\' – λ), (L' – L), (B + A) could be computed by any process, X', L' and B could be easily deduced therefrom. П For instance supposing X-λ = ^^, L'—L = AL, and B (π + A) ΔΑ, we shall have λ' = λ + Δ; L'L + AL; and B = ( + A) + ▲A. = The reason why these differential quantities AX, AL, ▲A, are computed in preference to X, L' and B, is that the former are susceptible of easier and more accurate deduction than the latter. On reference to pp. 161 and 169, of Col. Everest's account of the Indian arc published in 1847, it will be seen that the values of ▲, AL, and ▲▲ come out in infinite series. These series are rapidly convergent: Col. Everest uses only the first four terms and omits the others on account of their minuteness. For the purposes of this work, however, the first and second terms are all which will be required, the third and the fourth terms which are retained by Col. Everest being too minute to merit attention at this place. Limited to the 2nd term, the formula for the computation of latitudes, longitudes and azimuths as arranged by Col. Everest, are as follows: The terms P, Q, R, S, T...... which occur in these formulæ are composed of the numerical values of a and b given before, and of certain functions of the given latitude λ. Most of these terms are of tedious deduction, on which account it becomes necessary that they should be computed once for all, and registered under a Tabular Form, so as to be ready for use when required. Accordingly in the Great Trigonometrical Survey of India, we have the Table of P, Q, R, ...... computed for every 10' of latitude between the parallels of 8° and 35° - and we will give at the end of this Chapter an extract from this table, which will facilitate computations by Col. Everest's formulæ. The arrangement of this table is so simple that it hardly requires any explanation. Enter the table with the given latitude of station A. If λ is exactly found in the table, take out P, Q, R, ...... just as they stand in a line therewith. This is a very simple operation, but the exact agreement which we have supposed to exist between the given and the tabular latitude, seldom takes place in practice. In most instances the given latitude will lie between two tabular latitudes. In such cases take out P, Q, R, ...... appertaining to the next less tabular latitude and correct them in this wise. Take the difference between the given and the tabular latitude next less, and convert it to the denomination of a minute. The term so obtained being multiplied successively by the tabular differences for P, Q, R, S and T, and divided by 10, will furnish the required corrections, which will be negative in the case of P, R and Tand positive in that of Q and S. be P, Q, R, S, T being computed, other terms of the formulæ, such for instance as cos. A, tan. A, sec. λ sin. λ ..... may taken out from a common table of logarithms. When the terms dλ, d1L, ...... are computed, they will be in seconds and decimals thereof. The signs of these terms dependant upon the magnitude of the given azimuth A, may be easily taken out from the following table. After proper signs have been prefixed to dλ, d1L:—take the sums of λ and dλ:—of L and d2L:—and of d1A 1 and A. The three sums so obtained will be the values, the first 2 of AX, the second of AL, and the last of AA. Now A being applied to λ, AL to L, and AA to (π + A) the resulting elements will be λ', and L' and A'. By way of illustrating the computation of the latitudes, longitudes and azimuths of Trigonometrical Stations we subjoin the following example. Shevalingapah deduced from Yemshaw. |
