the Euler equations, transversality condition, Weierstrass condition and Clebsch condition are satisfied, and the second variation formed in the usual way from F and T) and the end functions is non-negative. Conjecture (S). In order that a smooth curve (1.9) shall give a strong proper relative minimum to the functional (1.5) in the class of curves satisfying (1.7) and (1.8), it is sufficient that the following condition be satisfied. To each nonidentically zero set [^'(x), £i, £>] satisfying

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... equations of variation of (1.7) and (1.8) there shall correspond multipliers X° = 0, X"(x) with which the Euler equation, transversality condition and strengthened Weierstrass and Clebsch conditions hold, and the second variation is positive.