Imagery around a skew ray

The one-one correspondence between rays in the object and image spaces of an optical instrument only yields a corresponding relation between points, even for a narrow ray pencil, in exceptional circumstances. The usual criterion by which conjugate points are defined fails unless neighbouring rays intersect in both spaces. In general this condition is not satisfied, and the criterion is therefore extended to include the points of nearest approach of non-intersecting rays. It is shown that pairs of conjugate points on a given chief ray determined according to this definition are coincident for a fan of near rays with one degree of freedom, that is the light path between the conjugate points is constant to the second order for the routes corresponding to all rays of this fan, and that the enlarged definition is applicable to all near rays. The relation connecting conjugate points is of the same form as for the simpler cases generally recognised, and may be determined by parallel projection through fixed points depending on the chief ray and on the particular fan under consideration. Apart from the exceptional case of complete anastigmatism due to the coincidence of the projective centres for all the fans, there can at most be two pairs of conjugate points free from astigmatism. Three conditions must be satisfied for stigmatic imagery at a given pair of points. The pencil around any skew ray has ten degrees of freedom, and the fundamental coefficients which determine the imagery completely are elements of a square matrix of the fourth order. Formulae are given for the construction of these matrices for refraction at each surface and for the transference from one surface to another. The matrix for the complete system is the product of the matrix factors for the successive elementary events taken in their proper order. Each refracting surface may be replaced by the osculating surface of the second degree at the point of refraction of the chief ray. It is assumed that the curvature of the surfaces is everywhere finite and continuous. A more natural matrix for refraction in space of three dimensions would be of the sixth order. Such matrices are derived from those of the fourth order. For theoretical investigations and perhaps also for routine numerical work, these matrices are preferable to those of lower order since they contain no quantities which are not relevant to the problem. The coefficients of the eikonal and of the characteristic function can be derived from the elements of either type of matrix. Fifteen independent relations between the coefficients of these matrices are obtained in two distinct forms.