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2n-2)-parameter family Amer arguments of f Bolza Calculus of Variations canonical differential equations Canonical Variables condition for problems conjugate points corners defining the transversal different from zero differentiate with respect Dubois-Reymond equations 2-8 Euler differential equations EULER EQUATIONS everywhere different existence theorems exists a set extremal arc family of extremals field form a fundamental function W(x fundamental system Furthermore G.A.Bliss Hamilton-Jacobi partial differential HAMILTON-JACOBI THEORY Hamiltonian Hence Henoe homogeneity property identically zero initial values Jacobi condition JACOBI EQUATIONS Jacobi's necessary condition Legendre Lemma linearly independent Math matrix McFerrin School minimizing arc Notations and formulae obtain original problem orthogonal variation parameter parametric form PARAMETRIC PROBLEMS partial differential equation path of integration Professor Bliss quadratic form region F satisfy the relation second variation Section solu solutions of differential SOLUTIONS OP JACOBI solvable sub-space system of orthogonal Tennessee Theorem XI Theorem XIII theorems for differential throughout F tions Trans University of Chicago