Posted by Sam on Thursday, May 27, 2010 at 12:13am.
The Brachistochrone (shortest sliding time) and the Tautochrone (equal sliding time wherever release occurs) are the same curve. Both are cycloids - loci of the a point at the edge of a wheel as it rolls. The situation is discussed in G. B. Thomas' classic textbook "Calculus and Analytic Geometry", 3rd edition, published by Addison Wesley in 1960. There is a ninth edition with the same title with Thomas and Finney listed as authors, published in 1999. I don't know if the proof is there. For a complete proof, Thomas refers to the book "Calculus of Variations" by G.A. Bliss, published in 1925. The original proof was by James and John Bernoulli.
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