  This paper develops an abstract framework for constructing ``seminormal
forms'' for cellular algebras. That is, given a cellular R-algebra A which is
equipped with a family of JM-elements we give a general technique for
constructing orthogonal bases for A, and for all of its irreducible
representations, when the JM-elements separate A. The seminormal forms for A
are defined over the field of fractions of R. Significantly, we show that the
Gram determinant of each irreducible A-module is equal to a product of certain
structure constants coming from the seminormal basis of A. In the non-separated
case we use our seminormal forms to give an explicit basis for a block
decomposition of A.
  The appendix, by Marcos Soriano, gives a general construction of a complete
set of orthogonal idempotents for an algera starting from a set of elements
which act on the algebra in an upper triangular fashion. The appendix shows
that constructions with "Jucys-Murphy elements"depend, ultimately, on the
Cayley-Hamilton theorem.
