By Giusti M., Weil J.-A., Zimmermann P.
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Additional resources for Calcul formel (Journes X-UPS 1997)
Which also is a vector-form, by the principles of the 20th article.
52. ) so that it is, in general, by the principles of articles 40, 41, 42, 43, the reflexion of the vector ι with respect to the vector τ , and that thus the direction of τ is exactly intermediate between the directions of ι and µ. ) may therefore represent the reflexion of the first cyclic normal ι, with respect to any reflecting line which is parallel to the vector τ , which latter vector is tangential to one of the curves of curvature on the ellipsoid. ), and suppressing the scalar factor τ 2 , we find this new equation: S .
For it shows that if a straight line µ be found, in any plane parallel to the given lines ν and κ, such that the bisector τ of the angle between this line µ and a line parallel to the other given line ι shall be perpendicular to the given line ν, then this bisecting line τ will have the sought direction of a tangent to a line of curvature. But it is possible to deduce a geometrical determination, or construction, more simple and direct than this, by carrying the calculation a little further. 53.