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2n-2)-parameter family Amer arguments of f Bolza Calculus of Variations canonical differential equations Canonical Variables class C1 condition for problems conjugate points constant defining the transversal different from zero equa Euler differential equations EULER EQUATIONS everywhere different existence theorems exists a set extremal arc family of extremals field form a fundamental function W(x functions x^(t fundamental system Furthermore G.A.Bliss Hamilton-Jacobi partial differential Hamilton-Jacobi theory Hamiltonian function Hence Henoe homogeneity property identically zero initial values Jacobi equations Jacobi's condition Jacobi's necessary condition Legendre Lemma linearly independent matrix 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 satisfies the Hamilton-Jacobi satisfy the relation second variation slope-functions pL(x solu solutions of differential sub-space Tennessee Theorem VII Theorem XIII theorems for differential throughout F tions Trans University of Chicago variations in parametric Weierstrass E-function