  The twistor space of self-dual positive Einstein manifolds naturally admits
two 1-parameter families of Riemannian metrics, one is the family of canonical
deformation metrics and the other is the family introduced by B. Chow and D.
Yang in 1989. The purpose of this paper is to compare these two families. In
particular we compare the Ricci tensor and the behavior under the Ricci flow of
these families. As an application, we propose a new proof to the fact that a
locally irreducible self-dual positive Einstein 4-manifold is isometric to
either $S^4$ with a standard metric or $\PP^2(\C)$ with a Fubini-Study metric.
