C1 r2 x r31 1 r1 x r31 c3 r2 x rx 1 r1 x r31

and, since | a x b | is twice the area of the triangle having a and b as two of its sides, it follows that c1 and c3 are the ratios of the areas of the triangles formed by the Sun and the respective positions of the body. Bouguer made the approximation c1 = (t3 - t2)/(t3 - h), c3 = (t2 - h)/(t3 - ¿1), which, due to Kepler's second law, is equivalent to replacing the areas of the relevant triangles by the areas of the corresponding sectors.

36 See, for example, Escobal (1965), Section 3.7.2.

Abhandlung uber die leichteste und bequemste Methode die Bahn eines Cometen zu berechnen. Brief extracts are translated in Shapley and Howarth (1929).

If sufficiently accurate values for c1 and c3 are known, the solution can then be obtained. Since ri = R + pi, Eqn (10.3) can be thought of as three scalar equations for the three unknowns |pi |. Once these geocentric distances have been determined, the heliocentric position vectors ri and, hence, the spatial coordinates of three points on the orbit, are known. Any two of these points then can be used to recover the orbital elements and the third point used as a check. The process then can be repeated until the required accuracy is obtained. However, Bouguer's approximate formulas for c1 and c3 are not sufficiently accurate in practice, since small changes in c1 and c3 can be shown to lead to large changes in the calculated heliocentric positions.

In the Theoria motus, Gauss explained very clearly the nature of the problem of determining an orbit from three latitude and longitude observations. He pointed out that such a procedure will be impractical if the orbit lies close to the ecliptic plane. This is because if the orbit were precisely in the ecliptic, then three longitude observations would not be sufficient to determine the four remaining elements (i and ^ being redundant). For orbits close to the ecliptic, it is better to use four longitude and two latitude data. Gauss treated this problem separately and we will not consider it further here.

For the three-observation problem, the observations that are used for the initial orbit determination should, Gauss said, not be too close (or the inevitable inaccuracies in the observations will affect the calculation too much) and not be separated too widely (or the resulting elements will be poor approximations). What is too close and too far apart must be judged from experience. Gauss claimed, however, that he had used his method successfully to determine the orbit of Juno from observations covering a heliocentric motion of 7° 35' and had computed the orbit of Ceres from observations embracing 62° 55' of arc.

It is possible, in principle, to reduce the six equations in six unknowns to a single equation for each of the elements, but Gauss noted that the complicated nature of the equations makes this impossible in practice. Instead, he did the next-best thing. There are many ways in which the problem can be reduced to two equations of the form X(P, Q) = 0 and Y (P, Q) = 0 in terms of two unknown quantities P and Q. These need not necessarily be two of the elements, but the elements must be calculated readily once P and Q have been found. The selection of P and Q is crucial to the success of the orbit determination process. First, one would like the functions X and Y to be as simple as possible; second, it is desirable that approximations to P and Q are calculated easily, and third, the numerical solution of the equations X = Y = 0 should be efficient and, in particular, as insensitive as possible to numerical error.

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