Some new methods of optical design developed from the properties of differential transfer coefficients.

The main attempts which have been made to develop systematic methods of optical design have resulted in various algebraic solutions of the theory of the aberrations of an optical system. The essential content of each of these is an analysis of the dependence of the aberrations on the curvatures, refractive indices, axial separation, and aperture etc., i.e. on the general parameters of the system. On account of the complexity of the the problem the common procedure has been to expand. various functions In series, the expansions being carried as far as terms of a selected order, with the result that approximations of various orders have been achieved. In addition to these algebraic theories there is a trigonometrical approach based on the ray trace which does not consider orders of aberrations but employs measures of the total aberrations which are present in the image formed by the system. In the general practice of optical designing the main use of this method has been as a final test of the correction of the system. Proponents of the algebraic methods have generally considered trigonometrical ray tracing as sterile and uninstructive. For example, H. D. Taylor (1906) says 'Although the trigonometrical calculation of the course of a ray through an optical system is often highly desirable yet these are merely mechanical processes, which more especially when applied to oblique and eccentric pencils, do not lend themselves at all to analysis. They are empirical and uninstructive, or at any rate barren of enlightenment unless a large number of calculations are carried out in which certain factors such as radii or separations are varied, and the results of such variations carefully noted. All this involves much empirical work, whereas, by the aid of algebraic formulae, although they may not be quite exact, leading principles can be established and the tendencies of the corrections consequent upon the variation of any one term can always be worked out with very little trouble, and it is by the intelligent grasp of the general tendencies that an aptical construction may be varied in its parts until the utmost possible perfection is realised.' Conrady (1929), however, views the algebraic and trigonometrical processes as complementary and mutually indispensable. In the preface to his treatise he speaks of 'the elegant but approximate algebraical methods' which furnish a rough solution, and 'the rigourously exact method of trignnometrical ray tracing (which) quickly and systematically adds the necessary finishing touches.' A little experience in optical design is sufficient to show that there is still much to be desired in the 'systematic' way in which the finishing touches are added to a rough design.