In [I, 2], representations of solutions of operator equations and systems in Hilbert space are obtained in the form of certain general continual integrals. The solution of a linear Fredholm integral equation of the second kind is expressed in terms of a Wiener integral in [3]. In what follows here, these results are generalized to the case of a system of Fredholm and Volterra integral equations of the second kind [4], which allows us to represent the solution of a system of partial differential equations in the form of multiple Wiener integrals [4-6]. In the present article, the concept of a Wiener integral in the space of continuous functions of infinitely many variables [7] is generalized to the case of a Cartesian product of these spaces (i.e., the multiple Wiener integral is considered), the transformation of this integral under linear changes of variable is studied, and solutions of systems of linear Fredholm and Volterra integral equations of the second kind, as well as of systems of partial differential equations are represented in terms of these multiple Wiener integrals. i. We introduce the concept of a multiple Wiener integral in the space of continuous functions of infinitely many variables, and we consider the question of its relationship to the multiple Wiener integral in the space of continuous functions of finitely many variables. Let Coo ~ be the space of real continuous functions q~(t), t = (t I, ..., t m, ...)6Qm = {0 ~ t k ~ i, k = i, -}, and let ~(/)It~=0=0, k----I~'~-, ~, be a cylindrical set in C| ~ [7]. We consider the Cartesian products Cn,~ ~ = Coo ~ x ... x Coo ~ and ~a.~ =~ x ...x ~|