ProtoQuant.com

Glaisher’s work (Glaisher, 1878) does nothing more but shows that all of the results in characterizing the forces, based on the second principle of dynamics, can be recovered from Newton’s essential propositions, by casting them into analytical form. And the basic proposition Glaisher chose is the Corollary 3 of the Proposition vii of Newton (Newton, 1995; p. 48):

The force by which the body P in any orbit revolves about the center of force S, is to the force by which the same body may revolve in the same orbit, and the same periodic time, about any other center of force R, as the solid SP´RP², contained under the distance of the body from the first center of forces, and the square of its distance from the second center of force R, to the cube of the right line SG, drawn from the first center of force S parallel to the distance RP of the body from the second center of force R, meeting the tangent PG of the orbit in G.

Referring to the attached figure, the equation expressing this Corollary is

(1)

Glaisher writes this in the form

(2)

Then by similitude this ratio can be written as

(3)

which is particularly adequate for writing in an analytical form. Choosing S as the origin of a reference frame, and referring the generic coordinates, x,h say, to this frame, the equation of the orbit is

(4)

Then the tangent to this conic in the current point P of coordinates x, y is

(5)

Thus one can write

(6)

whereis the vector of components

(7)

. The equation (3) can then be written in the form

(8)

and if we choose R as the center of the conic section, then

(9)

Here μ is a constant coming from the law of force towards center allowing for the same conic, which is known to be proportional with the distance. Using (7) in order to calculate the dot product from the parenthesis of equation (9) gives

(10)

which is the main of Glaisher’s results. It was also established earlier by Hamilton, which is why it is also known under his name.

References

Glaisher, J. W. L. (1878): On the Law of Force to any Point in the Plane of Motion, in order that the Orbit may be always a Conic, Monthly Notices of the Royal Astronomical Society, Vol. 39, pp. 77–91

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