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admissible arc admissible variation arbitrary arc E^2 arc length assume Calculus of Variations conjugate point consider continuous function converges corners defined dependent variables dt dx DuBois-Reymond envelope Euler's equation example exist extremal arc f dx family of extremals four parameter family G. A. Bliss geodesic implies independent solutions inflexion integral integrand interval Jacobi's differential equation Lagrange least of class lecture Legendre Legendre's condition Legendre's determinant limit maxima and minima Mayer field minimizing arc n'+f necessary condition notation osculating plane point Pq Proof proper minimum properties quadratic form reduces relations second variation slope solution of Euler's solved strong relative minimum subsidiary equations sufficient conditions Suppose surface tangent line Taylor series theory tremal variational calculus weak relative minimum Weierstrass xl x2 y y y z yy yy zero