Gravitational lensing and the maximum number of images

We show that in a configuration of point masses, replacing one of the point deflectors by a spherically symmetric distributed mass only introduces one extra image. This extra image is located inside the radius of the distributed mass added. We extend a result available for a set of n point masses, by showing that the maximum number of images that n distributed lensing objects located on a plane can create is N sub(max) = 6(n - 1) + 1 for n greater than or equal to 2. Our result is generic in that the mass density for each deflector is only constrained to diverge slower than 1/r as r approaches the center of the mass.