  More than 150 years after their invention by Hamilton, quaternions are now
widely used in the aerospace and computer animation industries to track the
paths of moving objects undergoing three-axis rotations. It is shown here that
they provide a natural way of selecting an appropriate ortho-normal frame --
designated the quaternion-frame -- for a particle in a Lagrangian flow, and of
obtaining the equations for its dynamics. How these ideas can be applied to the
three-dimensional Euler fluid equations is then considered. This work has a
bearing on the issue of whether the Euler equations develop a singularity in a
finite time. Some of the literature on this topic is reviewed, which includes
both the Beale-Kato-Majda theorem and associated work on the direction of
vorticity by both Constantin, Fefferman & Majda and Deng, Hou and Yu. It is
then shown how the quaternion formulation provides a further direction of
vorticity result using the Hessian of the pressure.
