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This paper studies the modal character of optical, dielectric waveguides with arbitrary spatial variation (in one dimension) of the dielectric coefficient, a subject vital to the development of optical fibers, strip waveguides, semiconductor lasers, etc. Within (but not limited to) this context, a new mathematical tool is presented for analysis of solutions of the general second-order, linear, homogeneous differential equation, which, when put into its canonical form, is the wave equation: . The central theme is the development and study of the uses of a "mantle function" which simultaneously describes two, complementary solutions. It is shown how the method provides new analytical insights; new, more convenient techniques for numerical evaluation of eigensolutions; and a firm basis for evaluating the merits of existing approximation methods, e.g., the Wentzel-Kramer-Brillouin (WKB) method.