These errors can nevertheless be almost entirely avoided by using the impersonal micrometer of Dr Repsold (Hamburg, 1889). In this device there is a movable micrometer wire which is brought by hand into coincidence with the star and moved along with it; at fixed points there are electrical contacts, which replace the fixed wires. Experiments at the Geodetic Institute and Central Bureau at Potsdam in 1891 gave the following personal equations in the case of four observers:—
| Older Procedure. | New Procedure. | |
| A − B | −0s.108 | −0s.004 |
| A − G | −0s.314 | −0s.035 |
| A − S | −0s.184 | −0s.027 |
| B − G | −0s.225 | +0s.013 |
| B − S | −0s.086 | −0s.023 |
| G − S | +0s.109 | −0s.006 |
These results show that in the later method the personal equation is small and not so variable; and consequently the repetition of longitude determinations with exchanged observers and apparatus entirely eliminates the constant errors, the probable error of such determinations on ten nights being scarcely ±0s.01.
Calculation of Triangulation.
The surface of Great Britain and Ireland is uniformly covered by triangulation, of which the sides are of various lengths from 10 to 111 miles. The largest triangle has one angle at Snowdon in Wales, another on Slieve Donard in Ireland, and a third at Scaw Fell in Cumberland; each side is over a hundred miles and the spherical excess is 64″. The more ordinary method of triangulation is, however, that of chains of triangles, in the direction of the meridian and perpendicular thereto. The principal triangulations of France, Spain, Austria and India are so arranged. Oblique chains of triangles are formed in Italy, Sweden and Norway, also in Germany and Russia, and in the United States. Chains are composed sometimes merely of consecutive plain triangles; sometimes, and more frequently in India, of combinations of triangles forming consecutive polygonal figures. In this method of triangulating, the sides of the triangles are generally from 20 to 30 miles in length—seldom exceeding 40.
The inevitable errors of observation, which are inseparable from all angular as well as other measurements, introduce a great difficulty into the calculation of the sides of a triangulation. Starting from a given base in order to get a required distance, it may generally be obtained in several different ways—that is, by using different sets of triangles. The results will certainly differ one from another, and probably no two will agree. The experience of the computer will then come to his aid, and enable him to say which is the most trustworthy result; but no experience or ability will carry him through a large network of triangles with anything like assurance. The only way to obtain trustworthy results is to employ the method of least squares. We cannot here give any illustration of this method as applied to general triangulation, for it is most laborious, even for the simplest cases.
Three stations, projected on the surface of the sea, give a spherical or spheroidal triangle according to the adoption of the sphere or the ellipsoid as the form of the surface. A spheroidal triangle differs from a spherical triangle, not only in that the curvatures of the sides are different one from another, but more especially in this that, while in the spherical triangle the normals to the surface at the angular points meet at the centre of the sphere, in the spheroidal triangle the normals at the angles A, B, C meet the axis of revolution of the spheroid in three different points, which we may designate α, β, γ respectively. Now the angle A of the triangle as measured by a theodolite is the inclination of the planes BAα and CAα, and the angle at B is that contained by the planes ABβ and CBβ. But the planes ABα and ABβ containing the line AB in common cut the surface in two distinct plane curves. In order, therefore, that a spheroidal triangle may be exactly defined, it is necessary that the nature of the lines joining the three vertices be stated. In a mathematical point of view the most natural definition is that the sides be geodetic or shortest lines. C. C. G. Andrae, of Copenhagen, has also shown that other lines give a less convenient computation.
K. F. Gauss, in his treatise, Disquisitiones generales circa superficies curvas, entered fully into the subject of geodetic (or geodesic) triangles, and investigated expressions for the angles of a geodetic triangle whose sides are given, not certainly finite expressions, but approximations inclusive of small quantities of the fourth order, the side of the triangle or its ratio to the radius of the nearly spherical surface being a small quantity of the first order. The terms of the fourth order, as given by Gauss for any surface in general, are very complicated even when the surface is a spheroid. If we retain small quantities of the second order only, and put A, B, C for the angles of the geodetic triangle, while A, B, C are those of a plane triangle having sides equal respectively to those of the geodetic triangle, then, σ being the area of the plane triangle and a, b, c the measures of curvature at the angular points,
| A = A + σ(2a + b + c) / 12, |
For the sphere a = b = c, and making this simplification, we obtain the theorem previously given by A. M. Legendre. With the terms of the fourth order, we have (after Andrae):
| A − A = | ε | + | σ | k ( | m2 − a2 | k + | a − k | ), |
| 3 | 3 | 20 | 4k | |||||
| B − B = | ε | + | σ | k ( | m2 − b2 | k + | b − k | ), |
| 3 | 3 | 20 | 4k | |||||
| C − C = | ε | + | σ | k ( | m2 − c2 | k + | c − k | ), |
| 3 | 3 | 20 | 4k |
in which ε = σk {1 + (m2k / 8)}, 3m2 = a2 + b2 + c2, 3k = a + b + c. For the ellipsoid of rotation the measure of curvature is equal to 1/ρn, ρ and n being the radii of curvature of the meridian and perpendicular.
It is rarely that the terms of the fourth order are required. As a rule spheroidal triangles are calculated as spherical (after Legendre), i.e. like plane triangles with a decrease of each angle of about ε /3; ε must, however, be calculated for each triangle separately with its mean measure of curvature k.
The geodetic line being the shortest that can be drawn on any surface between two given points, we may be conducted to its most important characteristics by the following considerations: let p, q be adjacent points on a curved surface; through s the middle point of the chord pq imagine a plane drawn perpendicular to pq, and let S be any point in the intersection of this plane with the surface; then pS + Sq is evidently least when sS is a minimum, which is when sS is a normal to the surface; hence it follows that of all plane curves on the surface joining p, q, when those points are indefinitely near to one another, that is the shortest which is made by the normal plane. That is to say, the osculating plane at any point of a geodetic line contains the normal to the surface at that point. Imagine now three points in space, A, B, C, such that AB = BC = c; let the direction cosines of AB be l, m, n, those of BC l ′, m′, n′, then x, y, z being the co-ordinates of B, those of A and C will be respectively—
| x − cl : y − cm : z − cn |
Hence the co-ordinates of the middle point M of AC are x + 12c(l ′ − l), y + 12c(m′ − m), z + 12c(n′ − n), and the direction cosines of BM are therefore proportional to l ′ − l : m′ − m : n′ − n. If the angle made by BC with AB be indefinitely small, the direction cosines of BM are as δl : δm : δn. Now if AB, BC be two contiguous elements of a geodetic, then BM must be a normal to the surface, and since δl, δm, δn are in this case represented by δ(dx/ds), δ(dy/ds), δ(dz/ds), and if the equation of the surface be u = 0, we have
| d 2x | / | du | = | d 2y | / | du | = | d 2z | / | du | , |
| ds2 | dx | ds2 | dy | ds2 | dz |
which, however, are equivalent to only one equation. In the case of the spheroid this equation becomes
| y | d 2x | − | d 2y | = 0, |
| ds2 | ds2 |
which integrated gives ydx − xdy = Cds. This again may be put in the form r sin a = C, where a is the azimuth of the geodetic at any point—the angle between its direction and that of the meridian—and r the distance of the point from the axis of revolution.
From this it may be shown that the azimuth at A of the geodetic joining AB is not the same as the astronomical azimuth at A of B or that determined by the vertical plane AαB. Generally speaking, the geodetic lies between the two plane section curves joining A and B which are formed by the two vertical planes, supposing these points not far apart. If, however, A and B are nearly in the same latitude, the geodetic may cross (between A and B) that plane curve which lies nearest the adjacent pole of the spheroid. The condition of crossing is this. Suppose that for a moment we drop the consideration of the earth’s non-sphericity, and draw a perpendicular from the pole C on AB, meeting it in S between A and B. Then A being that point which is nearest the pole, the geodetic will cross the plane curve if AS be between 14AB and 38AB. If AS lie between this last value and 12AB, the geodetic will lie wholly to the north of both plane curves, that is, supposing both points to be in the northern hemisphere.
The difference of the azimuths of the vertical section AB and of the geodetic AB, i.e. the astronomical and geodetic azimuths, is very small for all observable distances, being approximately:—
Geod. azimuth = Astr. azimuth − 112 e21 − e2 s2ρn (cos2 φ sin 2α + s4a|sin 2φ sin α), in which: e and a are the numerical eccentricity and semi-major axis respectively of the meridian ellipse, φ and α are the latitude and azimuth at A, s = AB, and ρ and n are the radii of curvature of the meridian and perpendicular at A. For s = 100 kilometres, only the first term is of moment; its value is 0″.028 cos2 φ sin 2α, and it lies well within the errors of observation. If we imagine the geodetic AB, it will generally trisect the angles between the vertical sections at A and B, so that the geodetic at A is near
