Vibrations of a Columnar Vortex

This is a case of fluid motion, in which the stream lines are approximately circles, with their centres in one line (the axis of the vortex) and the velocities approximately constant, and approximately equal at equal distances from the axis. As a preliminary to treating it, it is convenient to express the equations of motion of a homogeneous incompressible inviscid fluid (the description of fluid to which the present investigation is confined) in terms of “columnar co-ordinates” r, θ, z, that is co-ordinates such that r cos θ = x, r sin θ = y.