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We shall give here, in a slightly changed and heavily modernized notation, the essentials of Villarceau’s approach to characterizing the force responsible for the shape of the orbits of double stars. If the orbit of the companion is a conic in general position in a certain plane, whose equation is written in the form

(1)

The idea is to use the second principle of dynamics in the form (Whittaker, 1904). If the center of force is taken as origin of the reference frame, then the equations of motion can be written in the form

(2)

where f(x, y) is the magnitude of force, and a dot over the symbol means derivative with respect to time. As the force is central, the second Kepler law is valid, which can be written in the form

(3)

where stands for the “rate of area change” – a constant. On the other hand differentiating the equation of trajectory (1) one has

(4)

with C_x and C_y standing for the partial derivatives of the function C(x, y). Now one can solve the system of equations (3) and (4) for the components of velocity and then differentiate one of them with respect to time in order to obtain the value of force. First one has

(5)

and then by differentiation

(6)

Let’s evaluate the parenthesis in front of the fraction. Performing the derivatives in (1) gives the following table

(7)

After calculations, the parenthesis turns out to be

(8)

On the other hand, using the first two equations (7) gives

(9)

Now, gathering together the results (8) and (9) into equation (6) and comparing the result with the first of equations (2) gives the magnitude of the central force responsible for the orbit (1) in the form

(10)

Using here the equation (1) in order to eliminate the quadratic terms in the denominator gives

(11)

On the other hand, if the equation (1) is used to eliminate the linear terms from the denominator of (10) the result is

(12)

The equations (11) and (12) are the results obtained by Villarceau (Villarceau, 1850). Notice that, while the force is a central one, its magnitude does not depend exclusively on the distance: there is also a dependence on each coordinate separately, which is embodied in the shape and size of the orbit.

The known forces with magnitude depending only on the distance r can be obtained as particular cases of the above formulas. The isotropic elastic force, for instance, can be obtained directly from equation (11). When the center of force is the center of the orbit, one has a₁₃=a₂₃=0 and equation (11) becomes

(13)

which is the known result of Newton (Newton, 1995; p. 50, Proposition x. Problem v): if a material point describes a conic section under a centripetal force directed towards the center of that conic section, then the force is proportional to the distance between center and that point.

The forces inversely proportional with the square of distance are obtained when the center of force is located in one of the foci. In order to show this it is better to go back and consider the equation (1) for a conic written down in a way making explicit the fact that the focus is taken as center of force. That equation is

(14)

This expresses the fact that the eccentricity e is the ratio of the distances from the focus to the current point of the conic, which is r, and from the directrix to the current point of the conic, which is proportional to

For the sake of argument this quantity will be taken positive for now. Then, instead of (8) and (9) we will have the expressions

(15)

With these the equation (6) gives the following result of Newton ( Newton, 1995; p. 50, Proposition xi. Problem vi)

(16)

or, in words: if a material point describes an orbit which is a conic section under the action of a force directed towards a focus of that conic section, then the magnitude of that force is inversely proportional with the distance between focus and the point.

In perspective, Villarceau’s point is that the solar system is just a particular case of general natural systems, among which the binary stars can be seen just as another particular case. Fact is that Newton was well aware of the possibility of such general systems, as Glaisher’s arguments show. However, today, like one and a half century ago, this fact is completely forgotten, due to an advanced “mathematical technology”.

References

Newton, I. (1995): The Principia, Prometheus Books, Amherst, New York

Villarceau, Y. (1850): Memoires et Notes sur les Etoiles Doubles, Paris, Bachelier

Whittaker, E. T. (1904): Analytical Dynamics, Cambridge University Press