The localized-induction concept for the induction effect of a smooth curved vortex on itself is derived. This is an approximation applicable to the limiting case of a vortex filament of infinitesimal core size and of negligible long-distance effect, and was already successfully utilized in the investigations of the motion and deformation of a curved vortex filament given various initial configurations. Two theorems obtained under this concept are that the arc length of a vortex filament and the projected area of a closed vortex filament are both invariant with respect to time. These theoretical predictions are examined by a numerical analysis of the motion of an initially plane elliptic vortex ring of various eccentricities.
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Arms et al. (1965) studied this question.
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