  In this chapter of the book entitled, "Extending the Theory of Composites to
Other Areas of Science" [edited by Graeme W. Milton, 2016] we give a rigorous
derivation of the field equation recursion method in the abstract theory of
composites to two-component composites with isotropic phases. This method is of
great interest since it has proven to be a powerful tool in developing sharp
bounds for the effective tensor of a composite material. The reason is that the
effective tensor $\bf L_*$ can be interpreted in the general framework of the
abstract theory of composites as the $Z$-operator on a certain orthogonal
$Z(2)$ subspace collection. The base case of the recursion starts with an
orthogonal $Z(2)$ subspace collection on a Hilbert space $\cal H$, the
$Z$-problem, and the associated $Y$-problem. We provide some new conditions for
the solvability of both the $Z$-problem and the associated $Y$-problem. We also
give explicit representations of the associated $Z$-operator and $Y$-operator
and study their analytical properties. An iteration method is then developed
from a hierarchy of subspace collections and their associated operators which
leads to a continued fraction representation of the initial effective tensor
$\bf L_*$.
