Note on Tortuous Curves

The condition that the principal normals of one curve may also be the principal normals of a second curve is, as found by Bertrand, that a linear relation with constant coefficients should exist between the curvature and torsion of each curve. In seeking for pairs of curves such that the tangents, principal normals or binomials of one may be the tangents, principal normals or binormals of the other, there are six cases to be considered. The curves of Bertrand are furnished by one case, and a second case, that of evolutes and involutes, is also discussed in the text-books. Of the remaining four only one gives results worthy of mention. Bertrand's problem suggests the inquiry into the nature of the pair of curves when the binormal of one is the principal normal of the other. A certain quadratic relation of a simple character found to exist between the curvature and torsion of the second curve led me to a paper in the Comples Rendus of 1893, by Demoulin, in which the problem had been generalised. His method of solution is different, and no explicit results as to the nature of the curves are given in the paper. Since no indication of the discussion of the problem is given in the text-books I have seen, I venture to submit a note of some results.