Motion of a Vortex Filament and its Relation to Elastica

Steady rotation of a very thin plane vortex filament with uniform angular velocity \ (-\) in the direction opposite to the circulatory rotation around it is discussed on the basis of the localized induction equation. It is shown that the possible form of the filament is that of the plane elastic filament of flexural rigidity B under the action of thrust F applied at its ends provided that \ (G/=B/F\), where G is the coefficient of local induction proportional to the circulation of the filament.