a dandelion. This special type of vortex ring effectively stabilizes the seed as it travels through the air and increases the lift generated by the seed. Compared to a standard vortex ring, which is propelled downstream, the axially symmetric SVR remains attached to the pappus for the duration of its flight and uses drag to enhance the travel. These dandelion seed structures have been used to create tiny battery-free wireless sensors that can float in the wind and be dispersed across a large area. Theory Historical studies The formation of vortex rings has fascinated the scientific community for more than a century, starting with William Barton Rogers who made sounding observations of the formation process of air vortex rings in air, air rings in liquids, and liquid rings in liquids. In particular, William Barton Rogers made use of the simple experimental method of letting a drop of liquid fall on a free liquid surface; a falling colored drop of liquid, such as milk or dyed water, will inevitably form a vortex ring at the interface due to the surface tension. Vortex rings were first mathematically analyzed by the German physicist Hermann von Helmholtz, in his 1858 paper On Integrals of the Hydrodynamical Equations which Express Vortex-motion. Circular vortex lines For a single zero-thickness vortex ring, the vorticity is represented by a Dirac delta function as where denotes the coordinates of the vortex filament of strength in a constant half-plane. The Stokes stream function is:<Lamb></ref> with and where is the complete elliptic integral of the first kind and is the complete elliptic integral of the second kind. BEWARE that Mathematica decided on their own to change the argument of Elliptic Integrals to the SQUARE of \lambda) A circular vortex line is the limiting case of a thin vortex ring. Because there is no core thickness, the speed of the ring is infinite, as well as the kinetic energy. The hydrodynamic impulse can be expressed in term of the strength, or 'circulation' , of the vortex ring as . Thin-core vortex rings The discontinuity introduced by the Dirac delta function prevents the computation of the speed and the kinetic energy of a circular vortex line. It is however possible to estimate these quantities for a vortex ring having a finite small thickness. For a thin vortex ring, the core can be approximated by a disk of radius which is assumed to be