X. On the vibrations of a vortex ring, and the action of two vortex rings upon each other

Abstract

In the first part of the paper it is shown that if the circular axis of a vortex ring be displaced so as to be represented by the equations— ρ = a + α n cos nd , z = β u cos nd . when ρ is the distance of a point on the circular axis from the straight axis, and z the distance of a point on the circular axis from its mean plane, then— α n = A cos (ω e 2 /2 a 2 log 2 a / e n √ n 2 —1 . t + B), β n = A √ n 2 —1/ n sin ((ω e 2 /2 a 2 log 2 a / e n √ n 2 —1 . t + B), when ω is the angular velocity of molecular rotation, e the radius of the cross section of the vortex core, and a the radius of the aperture. The cross section is supposed small compared with the aperture so that e is small compared with a .