  In 1991, one of the authors showed the existence of quadratic transformations
between the Painleve' VI equations with local monodromy differences
$(1/2,a,b,\pm 1/2)$ and $(a,a,b,b)$. In the present paper we give concise forms
of these transformations. %, up to fractional-linear transformations. They are
related to the %better known quadratic transformations obtained by Manin and
Ramani-Grammaticos-Tamizhmani via Okamoto transformations. To avoid cumbersome
expressions with differentiation, we use contiguous relations instead of the
Okamoto transformations. The 1991 transformation is particularly important as
it can be realized as a quadratic-pull back transformation of isomonodromic
Fuchsian equations. The new formulas are illustrated by derivation of explicit
expressions for several complicated algebraic Painleve' VI functions.

Classical Analysis and ODEs