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A brief README note by Phil dated 1.22.17 describing the effort behind the tide potential method folder. He says the work became a Footnote at the end of Section 8.8. He knew Taylor and Butikov used the method but worked it out himself. The method gives a quartic surface very close to an ellipse, so the ellipse is not the only workable surface in the tide problem.
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README PhL 1.22.17
I spent an inordinate amount of time on this topic which has ended up as a Footnote section at the end of Section 8.8. I knew both Taylor anb Butikov used this method but I did not trace through either of their presentations, I wanted to do it myself. The key problem is that I was looking for an ellipse, when this method really gives a quartic which is very close to an ellipse. I think my Footnote is pretty concise considering all the complications. And it shows that the ellipse is not necessarily the only surface that you can work with in this tide problem.