co-ordinates, we can accurately calculate the corresponding astronomical
azimuths, i.e. those of the vertical section, and then proceed,
in the case of not too great distances, to determine the length and
the azimuth of the shortest lines. For any distances recourse must
again be made to Bessel’s formula.[1]
Let α, α′ be the mutual azimuths of two points A, B on a spheroid, k the chord line joining them, μ, μ′ the angles made by the chord with the normals at A and B, φ, φ′, ω their latitudes and difference of longitude, and (x2 + y2)/a2 + z2 b2 = 1 the equation of the surface; then if the plane xz passes through A the co-ordinates of A and B will be
| x = (a/Δ) cos φ, | x′ = (a/Δ′) cos φ′ cos ω, |
| y = 0 | y ′ = (a/Δ′) cos φ′ sin ω, |
| z = (a/Δ) (1 − e2) sin φ, | z′ = (a/Δ′) (1 − e2) sin φ′, |
where Δ = (1 − e2 sin2 φ)12, Δ′ = (1 − e2 sin2 φ′)12, and e is the eccentricity. Let f, g, h be the direction cosines of the normal to that plane which contains the normal at A and the point B, and whose inclinations to the meridian plane of A is = α; let also l, m, n and l′, m′, n′ be the direction cosines of the normal at A, and of the tangent to the surface at A which lies in the plane passing through B, then since the first line is perpendicular to each of the other two and to the chord k, whose direction cosines are proportional to x′ − x, y ′ − y, z′ − z, we have these three equations
| f (x′ − x) + gy ′ + h(z′ − z) = 0 |
| f l + gm + hn = 0 |
| f l′ + gm′ + hn′ = 0. |
Eliminate f, g, h from these equations, and substitute
| l = cos φ | l′ = − sin φ cos α |
| m = 0 | m′ = sin α |
| n = sin φ | n′ = cos φ cos α, |
and we get
(x′ − x) sin φ + y ′ cot α − (z′ − z) cos φ = 0.
The substitution of the values of x, z, x′, y ′, z′ in this equation will give immediately the value of cot α; and if we put ζ, ζ′ for the corresponding azimuths on a sphere, or on the supposition e = 0, the following relations exist
| cot α − cot ζ = e2 | cos φ Q |
| cos φ′ Δ |
| cot α′ − cot ζ′ = −e2 | cos φ′ Q | |
| cos φ Δ′ |
If from B we let fall a perpendicular on the meridian plane of A, and from A let fall a perpendicular on the meridian plane of B, then the following equations become geometrically evident:
| k sin μ sin α = (a/Δ′) cos φ′ sin ω |
| k sin μ′ sin α′ = (a/Δ) cos φ sin ω. |
Now in any surface u = 0 we have
| −cos μ = [ (x′ − x) | du | + (y ′ − y) | du | + (z′ − z) | du | ] / k ( | du2 | + | du2 | + | du2 | ) | 12 |
| dx | dy | dz | dx2 | dy2 | dz2 |
| cos μ′ = [ (x′ − x) | du | + (y ′ − y) | du | + (z′ − z) | du | ] / k ( | du2 | + | du2 | + | du2 | ) | 12 | . |
| dx′ | dy ′ | dz′ | dx′2 | dy ′2 | dz′2 |
In the present case, if we put
| 1 − | xx′ | − | zz′ | = U, |
| a2 | b2 |
then
| k2 | = 2U − e2 ( | z′ − z | ) | 2 |
| a2 | b |
Let u be such an angle that
| (1 − e2)12 sin φ = Δ sin u |
| cos φ = Δ cos u, |
then on expressing x, x′, z, z′ in terms of u and u ′,
U = 1 − cos u cos u ′ cos ω − sin u sin u ′;
also, if v be the third side of a spherical triangle, of which two sides are 12π − u and 12π − u ′ and the included angle ω, using a subsidiary angle ψ such that
we obtain finally the following equations:—
| k | = 2a cos ψ sin 12v |
| cos μ | = Δ sec ψ sin 12v |
| cos μ′ | = Δ′ sec ψ sin 12v |
| sin μ sin α | = (a/k) cos u ′ sin ω |
| sin μ′ sin α′ | = (a/k) cos u sin ω. |
These determine rigorously the distance, and the mutual zenith distances and azimuths, of any two points on a spheroid whose latitudes and difference of longitude are given.
By a series of reductions from the equations containing ζ, ζ′ it may be shown that
where φ0 is the mean of φ and φ′, and the higher powers of e are neglected. A short computation will show that the small quantity on the right-hand side of this equation cannot amount even to the thousandth part of a second for k < 0.1a, which is, practically speaking, zero; consequently the sum of the azimuths α + α′ on the spheroid is equal to the sum of the spherical azimuths, whence follows this very important theorem (known as Dalby’s theorem). If φ, φ′ be the latitudes of two points on the surface of a spheroid, ω their difference of longitude, α, α′ their reciprocal azimuths,
The computation of the geodetic from the astronomical azimuths has been given above. From k we can now compute the length s of the vertical section, and from this the shortest length. The difference of length of the geodetic line and either of the plane curves is
At least this is an approximate expression. Supposing s = 0.1a, this quantity would be less than one-hundredth of a millimetre. The line s is now to be calculated as a circular arc with a mean radius r along AB. If φ0 = 12 (φ + φ′), α0 = 12 (180° + α − α′), Δ0 = (1 − e2 sin2 φ0)12, then 1r = Δ0a (1 + e21 − e2 cos2 φ0 cos2 α0), and approximately sin (s/2r) = k/2r. These formulae give, in the case of k = 0.1a, values certain to eight logarithmic decimal places. An excellent series of formulae for the solution of the problem, to determine the azimuths, chord and distance along the surface from the geographical co-ordinates, was given in 1882 by Ch. M. Schols (Archives Néerlandaises, vol. xvii.).
In considering the effect of unequal distribution of matter in the earth’s crust on the form of the surface, we may simplify the matter by disregarding the considerations of rotation and eccentricity. In the first place, supposing the earth a sphere covered with a film of water, let the density ρ be a function of the distance from the centre so that surfaces of equal density are concentric spheres. Let now a disturbance of the arrangement of matter take place, so that the density is no longer to be expressed by ρ, a function of r only, but is expressed by ρ + ρ′, where ρ′ is a function of three co-ordinates θ, φ, r. Then ρ′ is the density of what may be designated disturbing matter; it is positive in some places and negative in others, and the whole quantity of matter whose density is ρ′ is zero. The previously spherical surface of the sea of radius a now takes a new form. Let P be a point on the disturbed surface, P′ the corresponding point vertically below it on the undisturbed surface, PP′ = N. The knowledge of N over the whole surface gives us the form of the disturbed or actual surface of the sea; it is an equipotential surface, and if V be the potential at P of the disturbing matter ρ′, M the mass of the earth (the attraction-constant is assumed equal to unity)
| M | + V = C = | M | − | M | N + V. |
| a + N | a | a2 |
As far as we know, N is always a very small quantity, and we have with sufficient approximation N = 3V/4πδa, where δ is the mean density of the earth. Thus we have the disturbance in elevation of the sea-level expressed in terms of the potential of the disturbing matter. If at any point P the value of N remain constant when we pass to any adjacent point, then the actual surface is there parallel to the ideal spherical surface; as a rule, however, the normal at P is inclined to that at P′, and astronomical observations have shown that this inclination, the deflection or deviation, amounting ordinarily to one or two seconds, may in some cases exceed 10″, or, as at the foot of the Himalayas, even 60″. By the expression “mathematical figure of the earth” we mean the surface of the sea produced in imagination so as to percolate the continents. We see then that the effect of the uneven distribution of matter in the crust of the earth is to produce small elevations and depressions on the mathematical surface which would be otherwise spheroidal. No geodesist can proceed far in his work without encountering the irregularities of the mathematical surface, and it is necessary that he should know how they affect his astronomical observations. The whole of this subject is dealt with in his usual elegant manner by Bessel in the Astronomische Nachrichten, Nos. 329, 330, 331, in a paper entitled “Ueber den Einfluss der Unregelmässigkeiten der Figur der Erde auf geodätische Arbeiten, &c.” But without entering into further details it is not difficult to see how local attraction at any station affects the determinations of latitude, longitude and azimuth there.
Let there be at the station an attraction to the north-east throwing the zenith to the south-west, so that it takes in the celestial sphere a position Z′, its undisturbed position being Z. Let the rectangular components of the displacement ZZ′ be ξ measured southwards
- ↑ Helmert, Theorien der höheren Geodäsie, 1. 232, 247.
