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accessory minimum problem admissible arc arc length assume Calculus of Variations conjugate point consider continuous function converges corners curve defined definition dense set dependent variable DuBois-Reymond dx y y en(x envelope Euler's equation everywhere example exist extremal solution family of extremals finite four parameter family geodesic implies independent solutions inflexion integral interval Jacobi's differential equation least of class Lecture Legendre Legendre's condition Legendre's determinant limit maxima and minima maximizing arc Mayer field minimizing arc n'+f necessary condition osculating plane partial derivatives path point of contact proof proper minimum properties quadratic form relations restricted neighbourhood satisfied second variation slope solution of Euler's solved strong relative minimum subsidiary equations sufficient conditions suitably restricted surface tangent line Taylor series theory tremal variational calculus weak relative minimum Weierstrass xl X2 yy yy yy yz yz yy zero