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In previous papers1'2-3 the author has developed a direct method of geometrical optics better adapted to the actual procedures of optical calculation than previous methods. In this paper we shall introduce in this direct theory Hamilton's characteristic function, and with it we shall find the image of an arbitrary surface in a given optical system with rotational symmetry. The author apologizes for changing his notations again. In the previous papers a ray was specified by the coordinates (x, y) of its intersection point with the object plane and by the optical direction cosines £> V< f = V/w2—£2—i?2, i.e., the direction cosines multiplied by the respective refractive indices. In this paper, x, y represent the optical lengths (i.e., lengths multiplied by the refractive index) of the coordinates of the intersection point with the coordinate plane, while £, t), f = \/1 — £2 — j?2 are the direction cosines. The reader may prove for himself that the fundamental formulas given in the former papers remain unaffected by this change in coordinates. 1. Hamilton's characteristic function and the direct method. In the previous papers we introduced the vector b with coordinates x, y, the vector t with coordinates £, t), and analogously the vectors b'(a:', y') and t'(£', 77') for the image ray, x', y' being the optical lengths of the coordinates of the intersection point of the image ray with the coordinate plane z' = 0, and 77' the direction cosines of this ray. We found the equation

Studies in optics. II. Analysis of a given system with the help of the characteristic function, using the direct method of analysis