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angle applied average axis calculation calculus of variations catenaries chain coefficients computation condition conformal map consider constant continued fraction converges coordinates corresponding cosh cubic curvature curve cuspidal cubic decimals denote density derivative determined differential equation dynamical system energy ensemble equal equilibrium extremals factors figures finite formula free end given gives Green's function group of order harmonic function Hence imaginary roots infinite number integral intersection interval limit logarithms Mathematics method minimum motion multiples multiply perfect negative obtained osculating plane partial differential equation particle plane player positive prime principal normal problem proof quantity region representative points respect result satisfy sinh solution stake statistical mechanics straight lines suppose surface surface of revolution symmetric symmetric group tangent theorem theory tion transversals vanish variable vertical x-axis zero