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By S. S. Chern

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8-6) also has (t, n, b) as its moving trihedron, K and T being its curvature and torsion, and s, because of Eq. (8-9), its arc length. Hence there exists one curve C with given K(s) and T(s) of which the moving trihedron at P(so) coincides with the coordinate axes. We now must show that every other curve C which can be brought into a one-to-one correspondence with C such that at corresponding points, given by equal s, the curvature and torsion are equal, is congruent to C. This means that C can be made to coincide with C by a motion in space.

Darboux, Lecons I, Ch. 2. We find Eq. (10-8) in G. Scheffers, Anwendung I, p. 298. EXAMPLES. Plane curve. When r = 0: df/f = -iK ds,(' f =ce'°, fl = e ", f2 = 0, =J Kds, f3 = 0, f4 = 1, which lead to the Eqs. (8-2) of the plane curve. Cylindrical helix. In this case the Riccati equation (10-4) can be written in the form (c constant) w' _ - ari(1 + 2cw - w2). Two integrals can immediately be found by taking w2 - 2cw + 1 = 0. The general solution of this equation can now be found by means of one quadrature.

For curves of constant slope the ratio of curvature to torsion is constant. Conversely, if for a regular curve this condition is satisfied, then we can always find a constant angle a such that n(K cos a - r sin a) = 0, ds (t cos a + b sin a) = 0, t FIG. 1-28 FIG. 1-29 34 CURVES [CH. 1 or t cos a + b sin a = a, constant unit vector, along the axis. Hence: cos a = a t. The curve is therefore of constant slope. We can express this result as follows: A necessary and sufficient condition that a curve be of constant slope is that the ratio of curvature to torsion be constant.

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