  We show the modular properties of the multiple 'elliptic' gamma functions,
which are an extension of those of the theta function and the elliptic gamma
function. The modular property of the theta function is known as Jacobi's
transformation, and that of the elliptic gamma function was provided by Felder
and Varchenko. In this paper we deal with the multiple sine functions, since
the modular properties of the multiple elliptic gamma functions result from the
equivalence between two ways to represent the multiple sine functions as
infinite product.
  We also derive integral representations of the multiple sine functions and
the multiple elliptic gamma functions. We introduce correspondences between the
multiple elliptic gamma functions and the multiple sine functions.
