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In this paper, we study surfaces in Euclidean 3-space that satisfy a Weingarten condition of linear type as κ 1 = mκ 2 + n, where m and n are real numbers and κ 1 and κ 2 denote the principal curvatures at each point of the surface. We investigate the existence of such surfaces parametrized by a uniparametric family of circles. We prove that the only surfaces that exist are surfaces of revolution and the classical examples of minimal surfaces discovered by Riemann. The latter situation onlydoi:10.1215/kjm/1250281603 fatcat:7g6cldby6jdovbqxi7e5k247eq