I had been reading up on the Brachistochrone problem, and what interested me was that Bernoulli actually put a second problem in his New Year's Day Programma. The Brachistochrone takes all the glory, but the following problem seems interesting:
``To find a curve such that the sum of the two segments $PK$ and $PL$, on a line drawn at random from a point $P$ to cut the curve in two points $K$ and $L$, though the two segments be raised to any power, is constant."
First, I am trying to even figure out what this is saying. So it seems necessary that the random line that is drawn through $P$ must intersect the curve at exactly 2 places, otherwise there's some ambiguity.
Second, does the constant they speak of vary depending on the point $P$? This also seems necessary, but I could be confused based on the curve at hand.
Third, assuming one understands the question, does anyone have a solution?
Any helpful hints, comments, or clarifications would be most helpful in allowing me to satisfy my curiosity. Thanks!
