the vertical section AB, and at B near the section BA.[1] The greatest distance of the vertical sections one from another is e2s3 cos2 φ0 sin 2α0/16a2, in which φ0 and α0 are the mean latitude and azimuth respectively of the middle point of AB. For the value s = 64 kilometres, the maximum distance is 3 mm.
An idea of the course of a longer geodetic line may be gathered from the following example. Let the line be that joining Cadiz and St Petersburg, whose approximate positions are—
| Cadiz. | St Petersburg. |
| Lat. 36° 22′ N. | 59° 56′ N. |
| Long.6°18′ W. | 30° 17′ E. |
If G be the point on the geodetic corresponding to F on that one of the plane curves which contains the normal at Cadiz (by “corresponding” we mean that F and G are on a meridian) then G is to the north of F; at a quarter of the whole distance from Cadiz GF is 458 ft., at half the distance it is 637 ft., and at three-quarters it is 473 ft. The azimuth of the geodetic at Cadiz differs 20″ from that of the vertical plane, which is the astronomical azimuth.
The azimuth of a geodetic line cannot be observed, so that the line does not enter of necessity into practical geodesy, although many formulae connected with its use are of great simplicity and elegance. The geodetic line has always held a more important place in the science of geodesy among the mathematicians of France, Germany and Russia than has been assigned to it in the operations of the English and Indian triangulations. Although the observed angles of a triangulation are not geodetic angles, yet in the calculation of the distance and reciprocal bearings of two points which are far apart, and are connected by a long chain of triangles, we may fall upon the geodetic line in this manner:—
If A, Z be the points, then to start the calculation from A, we obtain by some preliminary calculation the approximate azimuth of Z, or the angle made by the direction of Z with the side AB or AC of the first triangle. Let P1 be the point where this line intersects BC; then, to find P2, where the line cuts the next triangle side CD, we make the angle BP1P2 such that BP1P2 + BP1A = 180°. This fixes P2, and P3 is fixed by a repetition of the same process; so for P4, P5 .... Now it is clear that the points P1, P2, P3 so computed are those which would be actually fixed by an observer with a theodolite, proceeding in the following manner. Having set the instrument up at A, and turned the telescope in the direction of the computed bearing, an assistant places a mark P1 on the line BC, adjusting it till bisected by the cross-hairs of the telescope at A. The theodolite is then placed over P1, and the telescope turned to A; the horizontal circle is then moved through 180°. The assistant then places a mark P2 on the line CD, so as to be bisected by the telescope, which is then moved to P2, and in the same manner P3 is fixed. Now it is clear that the series of points P1, P2, P3 approaches to the geodetic line, for the plane of any two consecutive elements Pn−1 Pn, Pn Pn+1 contains the normal at Pn.
If the objection be raised that not the geodetic azimuths but the astronomical azimuths are observed, it is necessary to consider that the observed vertical sections do not correspond to points on the sea-level but to elevated points. Since the normals of the ellipsoid of rotation do not in general intersect, there consequently arises an influence of the height on the azimuth. In the case of the measurement of the azimuth from A to B, the instrument is set to a point A′ over the surface of the ellipsoid (the sea-level), and it is then adjusted to a point B′, also over the surface, say at a height h′. The vertical plane containing A′ and B′ also contains A but not B: it must therefore be rotated through a small azimuth in order to contain B. The correction amounts approximately to −e2h′ cos2φ sin 2α/2a; in the case of h′ = 1000 m., its value is 0″.108 cos2φ sin 2α.
This correction is therefore of greater importance in the case of observed azimuths and horizontal angles than in the previously considered case of the astronomical and the geodetic azimuths. The observed azimuths and horizontal angles must therefore also be corrected in the case, where it is required to dispense with geodetic lines.
When the angles of a triangulation have been adjusted by the method of least squares, and the sides are calculated, the next process is to calculate the latitudes and longitudes of all the stations starting from one given point. The calculated latitudes, longitudes and azimuths, which are designated geodetic latitudes, longitudes and azimuths, are not to be confounded with the observed latitudes, longitudes and azimuths, for these last are subject to somewhat large errors. Supposing the latitudes of a number of stations in the triangulation to be observed, practically the mean of these determines the position in latitude of the network, taken as a whole. So the orientation or general azimuth of the whole is inferred from all the azimuth observations. The triangulation is then supposed to be projected on a spheroid of given elements, representing as nearly as one knows the real figure of the earth. Then, taking the latitude of one point and the direction of the meridian there as given—obtained, namely, from the astronomical observations there—one can compute the latitudes of all the other points with any degree of precision that may be considered desirable. It is necessary to employ for this purpose formulae which will give results true even for the longest distances to the second place of decimals of seconds, otherwise there will arise an accumulation of errors from imperfect calculation which should always be avoided. For very long distances, eight places of decimals should be employed in logarithmic calculations; if seven places only are available very great care will be required to keep the last place true. Now let φ, φ′ be the latitudes of two stations A and B; α, α* their mutual azimuths counted from north by east continuously from 0° to 360°; ω their difference of longitude measured from west to east; and s the distance AB.
First compute a latitude φ1 by means of the formula φ1 = φ + (s cos α)/ρ, where ρ is the radius of curvature of the meridian at the latitude φ; this will require but four places of logarithms. Then, in the first two of the following, five places are sufficient—
| ε = | s2 | sin α cos α, η = | s2 | sin2 α tan φ1, |
| 2ρn | 2ρn |
| φ′ − φ = | s | cos (α − 23ε ) − η, |
| ρ0 |
| ω = | s sin (α − 13ε ) | , |
| n cos (φ′ + 13η) |
α* − α = ω sin (φ′ + 23η) − ε + 180°.
Here n is the normal or radius of curvature perpendicular to the meridian; both n and ρ correspond to latitude φ1, and ρ0 to latitude 12(φ + φ′). For calculations of latitude and longitude, tables of the logarithmic values of ρ sin 1″, n sin 1″, and 2nρ sin 1″ are necessary. The following table contains these logarithms for every ten minutes of latitude from 52° to 53° computed with the elements a = 20926060 and a : b = 295 : 294 :—
| Lat. | Log. 1ρsin 1″. | Log. 1n sin 1″. | Log. 12ρn sin 1″. |
| ° ′ | |||
| 520 | 7.9939434 | 7.9928231 | 0.37131 |
| 10 | 9309 | 8190 | 29 |
| 20 | 9185 | 8148 | 28 |
| 30 | 9060 | 8107 | 26 |
| 40 | 8936 | 8065 | 24 |
| 50 | 8812 | 8024 | 23 |
| 530 | 8688 | 7982 | 22 |
The logarithm in the last column is that required also for the calculation of spherical excesses, the spherical excess of a triangle being expressed by ab sin C/(2ρn) sin 1″.
It is frequently necessary to obtain the co-ordinates of one point with reference to another point; that is, let a perpendicular arc be drawn from B to the meridian of A meeting it in P, then, α being the azimuth of B at A, the co-ordinates of B with reference to A are
AP = s cos (α − 23ε), BP = s sin (α − 13ε),
where ε is the spherical excess of APB, viz. s2 sin α cos α multiplied by the quantity whose logarithm is in the fourth column of the above table.
If it be necessary to determine the geographical latitude and longitude as well as the azimuths to a greater degree of accuracy than is given by the above formulae, we make use of the following formula: given the latitude φ of A, and the azimuth α and the distance s of B, to determine the latitude φ′ and longitude ω of B, and the back azimuth α′. Here it is understood that α′ is symmetrical to α, so that α* + α′ = 360°.
Let
θ = sΔ/α, where Δ = (1 − e2 sin2 φ)12
and
| ζ = | e2 θ2 | cos2 φ sin 2α, ζ′ = | e2 θ3 | cos2 φ cos2 α; |
| 4 (1 − e2) | 6 (1 − e2) |
ζ, ζ′ are always very minute quantities even for the longest distances; then, putting κ = 90° − φ,
| tan | α′ + ζ − ω | = | sin 12(κ − θ − ζ′) | cot | α |
| 2 | sin 12(κ + θ + ζ′) | 2 |
| tan | α′ + ζ − ω | = | cos 12(κ − θ − ζ′) | cot | α |
| 2 | cos 12(κ + θ + ζ′) | 2 |
| φ′ − φ = | s sin 12(α′ + ζ − α) | ( 1 + | θ2 | cos2 | α′ − α | ); |
| ρ0 sin 12(α′ + ζ + α) | 12 | 2 |
here ρ0 is the radius of curvature of the meridian for the mean latitude 12(φ + φ′). These formulae are approximate only, but they are sufficiently precise even for very long distances.
For lines of any length the formulae of F. W. Bessel (Astr. Nach., 1823, iv. 241) are suitable.
If the two points A and B be defined by their geographical
- ↑ See a paper “On the Course of Geodetic Lines on the Earth’s Surface” in the Phil. Mag. 1870; Helmert, Theorien der höheren Geodäsie, 1. 321.
