One of the most powerful methods for bounding the effective tensors of composites is what has become known as the compensated compactness method or translation method. The method was introduced by Murat and Tartar (Tartar 1979b; Murat and Tartar 1985; Tartar 1985); see in particular theorem 8 of Tartar (1979b), and independently by Lurie and Cherkaev (1982, 1984a). While embodying many of the same ideas, there is a difference between their approaches. For nonlinear media the two approaches give different types of bounds, as discussed in section 25.1 on page 529: the compensated compactness method of Murat and Tartar gives bounds on the average fields, while the approach of Lurie and Cherkaev gives bounds on the energy. Since both approaches yield identical results for linear media, the term translation method [introduced in Milton (1990b)] will be used to encompass both. The name arises because the bounds can be obtained by shifting, that is, translating, the tensor field by a constant tensor and applying the classical bounds. [This approach has the advantage that by applying the same translation, but replacing the classical bounds by tighter correlation function dependent bounds, one generates improved bounds that include more detailed information about the composite microgeometry; see section 26.5 on page 560. It has the disadvantage that one does not see why it is natural to consider bounds on the translated medium in the first place; see section 25.1 on page 529.]