Consider the axisymmetric flow in cylindrical coordinate system  with velocity components  and vorticity components . Since  in axisymmetric flows, the vorticity components are 
Continuity equation allows to define a stream function  such that
(Note that the vorticity components  and  are related to  in exactly the same way that  and  are related to ). Therefore the azimuthal component of vorticity becomes
The inviscid momentum equations , where  is the Bernoulli constant,  is the fluid pressure and  is the fluid density, when written for the axisymmetric flow field, becomes
in which the second equation may also be written as , where  is the material derivative. This implies that the circulation  round a material curve in the form of a circle centered on -axis is constant.
If the fluid motion is steady, the fluid particle moves along a streamline, in other words, it moves on the surface given by constant. It follows then that  and , where . Therefore the radial and the azimuthal component of vorticity are
The components of  and  are locally parallel. The above expressions can be substituted into either the radial or axial momentum equations (after removing the time derivative term) to solve for . For instance, substituting the above expression for  into the axial momentum equation leads to
But  can be expressed in terms of  as shown at the beginning of this derivation. When  is expressed in terms of , we get
This completes the required derivation.