Hence in any infinitesimal part of the fluid the circulation is zero round every small plane curve passing through the vortex line; and consequently the circulation round any curve drawn on the surface of a vortex filament is zero.
The circulation being always zero round a small plane curve passing through the axis of spin in vortical motion, it follows conversely that a vortex filament is composed always of the same fluid particles; and since the circulation round a cross-section of a vortex filament is constant, not changing with the time, it follows from the previous kinematical theorem that aw is constant for all time, and the same for every cross-section of the vortex filament.
We consider in the first instance, and chiefly, a plane curve described according to a law.
Analogous to the order and class of a plane curve we have the order, rank and class of the system (assumed to be a geometrical one), viz.
The system has singularities, and there exist between m, r, is and the numbers of the several singularities equations analogous to Pliicker's equations for a plane curve.
