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The everyday meaning of the word "force" is so vague, t h a t it can scarcely be made to yield a satisfactory general intuition. Doubtless both Newton and Kepler remembered this everyday concept when they spoke of physical force. Indeed, this analogy is still plausible, even for us. I n order to make it fruitful in celestial mechanics, we shall abstract from this everyday concept a series of postulates which must hold if any idea of force is to be scientifically useful. I n what follows, we shall speak of force only in terms of these postulates.

E sin ω = a(l — e2) — r, t h a t is ξ . e cos ω -\- η . e sin ω = α(1 — e2) — r. 4(2) the areal velocity c is given by therefore c = TTV(1 — e2) . „_„_£)■ and c is also given by c = \rH = \{ξή - ηξ) = \ | r x v |. The rest of the calculation is as before: the unknowns e cos co and e sin ω are evaluated from the linear equations (1) and (2) etc. 3. The problems just discussed have significance in connexion with the practical problem of orbit determination; this is: from a sufficient number of observations of the directions of a planet, as seen from the Earth, and of the corresponding times, to determine the six elements of its orbit.

The variety of such vectors may be reduced by adding further plausible postulates, until only one vector satisfies all the postulates. This one will be defined as the force vector. For Kepler motion the velocity vector does not satisfy the second postulate and it may, therefore, be eliminated immediately. The vector of relative acceleration, however, fulfils all three postulates, as we will now show. 6(5)) it follows, by differentiating with respect to the time variable t, that r = a[—i sin E +JV(l ~ e*) c o s ■#)-#> where dots denote differentiation with respect to the time variable.

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