That’s a nice continuation of the story!

There’s a generalization of Marden’s theorem for polynomials of *higher* degree, and even rational functions, here:

• Ben-Zion Linfield, On the relation of the roots and poles of a rational function to the roots of its derivative, *Bull. Amer. Math. Soc.* **27** (1920), 17–21.

Nice! For anyone who doesn't click on all the pictures (or perhaps has trouble following the Spanish), there is one fact here that's not in John's post: the root of the second derivative is the centre of the ellipse. (It should be well known that the root of the derivative of a quadratic polynomial is the midpoint between the roots of the quadratic, so the root of the second derivative of a cubic is the midpoint between the roots of the first derivative. But it's still nice to see how this fits into the overall picture.)

]]>In 3 dimensional space, a line mass gives rise to a potential proportional to log(r) ,which is what we need for this part of the proof to work (examine it). In 3 dimensional space a point mass gives rise to a potential proportional to -1/r, and that wouldn’t work here.

In 2 dimensional space, a point mass gives rise to a potential proportional to log(r). However, we don’t get elliptical orbits from the gravitational field of a point mass in 2 dimensions—only in 3 dimensions.

So, I needed to use a carefully chosen combination of point masses and line masses in 3 dimensions, to get everything to work. The fact that it works at all is sort of exciting.

]]>(click on the arrows)

]]>Mathematics is full of things like this, which, which I find fascinating.

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