E42 -- De linea celerrimi descensus in medio quocunque resistente

(On the curve of fastest descent in whatever resistent medium)

Summary: Euler considers a variational approach to determine the various properties of the brachistochrone curve subject to friction, taking the form of \(\frac{v^n}{c^n}\), where \(v\) is the squared velocity and \(c\) is the squared velocity at which the forces of friction and gravity cancel. He then derives the brachistochrone property \(\frac{2v}{r} = N\), where \(r\) is the radius of curvature and \(N\) is the component of force that is perpendicular to the curve.

Euler recalls that the frictionless solution reads \(s^2 = 2a(s-x)\), for some constant \(a\), which is the already-established equation for a cycloid. He furthermore considers the frictionless case with a centripetal force taking the form \(y^m\), where \(y\) is the distance to the attracting point. If \(a\) is the initial distance from this point, the solution reads \(Az^2 = a^{m+3} - y^{m+3}\) for some constant \(A\).

Now, looking at the brachistochrone with friction, Euler calls the horizontal coordinate \(y\) and the vertical coordinate \(x\). Defining \(p = \frac{ds}{dx}\), Euler finds that the following equation must always hold:
\(\frac{1}{p^{3n}q^n} = \frac{ng^{n - 1}}{2^{n - 1}c^n}\int \frac{\left(p^2 - 1\right)^{n - 1}dp}{p^{2n}} = P^{-n},\)

which conveniently reduces for \(n=1\). He then finds the solution \(s = c\cdot \ln\left(\frac{s - ax - ac + c}{c - ac}\right)\), after which he proceeds to look for some expressions for the involved arc length. He ends by looking at how other methods would have influenced the solution and how this solution relates to the tautochrone.

According to the records, it was presented to the St. Petersburg Academy on February 4, 1734. Euler himself mentions that he read this treatise in the conference, in his letter to Daniel Bernoulli on February 16, 1734 (Bibl. math. 73, 1906/7, p. 136).

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