The case of two parallel infinite dielectric cylinders, of equal diameter and dielectric constant, illuminated by a plane electromagnetic wave is studied. The scattered electric and magnetic fields from both cylinders can be expressed as infinite series involving Hankel functions, and the coefficients of both satisfy coupled systems (which involve Bessel and Hankel functions). It is shown how these systems can be decoupled, and then, in turn, solved explicitly by successive approximations. Under explicit conditions on the physical parameters of the problem, uniform convergence of the successive approximation series is rigorously proven. This is achieved by utilizing certain bounds on Bessel and Hankel functions, which are derived from asymptotic and integral representations.