The method applied in the following paper to the discussion of the second variation in the case in which one end-point is movable on a fixed curve, is closely analogous to that of Weierstrass f in his treatment of the problem for fixed end-points. The difference arises from the fact that in the present case terms outside of the integral sign must be taken into consideration. As a result of the discussion the analogue of Jacobi's criterion will be derived, defining apparently in a new way the

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... tical point ^ for the fixed curve along which the endpoint varies. The relation between the critical and conjugate points is discussed in § 4. § 1. The expression for the variation of the integral. Consider a fixed curve D, x=f(u), y = g(u), and a fixed point B (xx, yx). Let C be a curve, x = <p(t), y = f(t), cutting D at A (u = uQ, t = t0), passing through B(t = tx), and making the integral