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Pascal's ring is a hexagon each of whose corners represents one of the six cardinal points of an optical system and whose sides represent relationships of relative axial position of the cardinal points. Changes to the ring represent the axial displacements of the cardinal points of the visual optical system of an eye that are caused when a spectacle lens compensates for the ametropia. Pascal's schema was described some 70 years ago with little theoretical justification. The purpose of this paper is to derive expressions for the axial locations of the cardinal points of a compound system consisting of an optical instrument and a visual optical system and for the shift caused by the instrument, and to provide theoretical justification for Pascal's schema. The cardinal points are treated not as separate entities but in a unified manner as special cases of an infinite class of special points. Expressions are derived using Gaussian optics. The results are specialized for the case of the eye's ametropia compensated by optical instruments in general and by spectacle lenses in particular. Pascal's schema is shown to be broadly correct although some modification is necessary for the effects on the incident cardinal points especially for the myopic eye., (Copyright © 2011 Elsevier Ltd. All rights reserved.)

Background: The pupil admits to the back of the eye only some of the light arriving from a point in space. As a result only a portion of the cornea is involved when an eye views the point; it is the effective corneal patch for that point. The location, size and shape of the patch are of interest for corneal refractive surgery inter alia. Previous studies have used geometrical optics and a simple model eye (a naked eye with a spherical, single-surface, centred cornea and a concentric circular pupil). Even for the simplest situations geometrical optics provides only implicit formulae which give little insight into relationships and require numerical solution., Purpose: To show how linear optics leads to explicit formulae that estimate the geometry of the effective corneal patch in a very wide range of situations. The eye is not restricted to a single refracting surface; the surfaces may be astigmatic and decentred or tilted and the pupil may be decentred and elliptical. The eye may contain implants and it may be looking through a spectacle lens or other optical instrument which may also contain astigmatic and decentred surfaces., Methods: Linear optics is used to provide general formulae for the geometry of the corneal patch. An appendix illustrates application to some particular cases., Results: General formulae are obtained for the location and geometry of the effective corneal patch for object points that may be near or distant. Formulae are presented in particular for the special case of the naked eye and the case in which all surfaces are spherical and centred on a common axis. Numerical examples in the appendix allow comparison of results obtained via geometrical and linear optics., Conclusions: In using linear optics one sacrifices some accuracy at increasing angles away from the longitudinal axis but there is considerable gain in the complexity and range of problems that can be tackled, and the explicit formula one obtains clearly exhibit relationships among parameters of clinical relevance., (© 2010 The College of Optometrists.)

The routine of subjective refraction is usually understood, explained and taught in terms of the relative positions of line or point foci and the retina. This paper argues that such an approach makes unnecessary and sometimes invalid assumptions about what is actually happening inside the eye. The only assumption necessary in fact is that the subject is able to guide the refractionist to (or close to) the optimum power for refractive compensation. The routine works even in eyes in which the interval of Sturm does not behave as supposed; it would work, in fact, regardless of the structure of the eye. The idealized subjective refraction routine consists of two steps: the first finds the best sphere (the stigmatic component) and the second finds the remaining Jackson cross-cylinder (the antistigmatic component). The model makes use of the concept of symmetric dioptric power space. The second part of the refraction routine can be performed with Jackson cross-cylinders alone. However, it is usually taught and practiced using spheres, cylinders and Jackson cross-cylinders in a procedure that is not easy to understand and learn. Recognizing that this part of the routine is equivalent to one involving Jackson cross-cylinders only allows one to teach and understand the procedure more naturally and easily.

Round-off of spherocylindrical powers, to multiples of 0.25 D (for example) in the case of sphere and cylinder, and 1 or 5 degrees in the case of axis, represents a type of distortion of the data. The result can be artefacts in graphical representations, which can mislead the researcher. Lines and clusters can appear, some caused by moiré effects, which have no deeper significance. Furthermore artefacts can obscure meaningful information in the data including bimodality and other forms of departure from normality. A process called unrounding is described which largely eliminates these artefacts; each rounded power is replaced by a power chosen randomly from the powers that make up what is called the error cell of the rounded power.

For statistical and other purposes one needs to be able to determine an average eye. An average of refractive errors is readily calculated as an average of dioptric power matrices. A refractive error, however, is not so much a property of the eye as a property of the compensating lens in front of the eye. As such, it ignores other aspects of the optical character of the eye. This paper discusses the difficulties of finding a suitable average that fully accounts for the first-order optics of a set of optical systems. It proposes an average based on ray transferences and logarithms and exponentials of matrices. Application to eyes in particular is discussed.

In Gaussian optics properties such as dioptric power, lateral and angular magnification and thickness are simple scalar concepts. In linear optics, the optics of thick astigmatic systems, however, these concepts generalize to three-dimensional concepts in some cases (the dioptric power of thin systems, for example) and to four-dimensional concepts in general. As a result, the quantitative treatment of these properties in astigmatic systems presents challenges to the researcher in optometry, ophthalmology, and vision science. Considerable progress has been made only in the case of dioptric power. This paper presents a generalized approach to astigmatic optics which allows different physical properties to be treated in the same way: the theory is unified and, in a sense, complete. Mathematical and statistical methods developed for treating one concept become directly applicable to others. The paraxial optical properties of any optical system are completely defined by the 4 x 4 ray transfer matrix, called here the (ray) transference. The transference defines four fundamental properties of an optical system, tentatively called here positional magnification, optical thickness, divergence, and directional magnification. They are the four 2 x 2 submatrices A, B, C, and D of the transference. Each fundamental property is a modification of a familiar concept. Divergence is the negative of dioptric power expressed as the dioptric power matrix F. The four fundamental optical properties A, B, C, and D, and the derived property F, despite being different physically, all have the same underlying mathematical structure. This fact is exploited in developing a unified theory. The theory is complete in the sense that the fundamental properties fully characterize the paraxial optics of any system. The paper presents a general treatment that applies to any of the five properties. The implications are far reaching and extend beyond what can be described in the paper. Dioptric power of thin systems is treated as a particular application of the general theory. The result is the resolution of a number of issues of current interest to the researcher. It is shown, for example, that root-mean-squared (curvital) power, root-mean-squared torsional power, and length of the power vector (or dioptric strength) have a Pythagorean relationship, the power vector being the hypotenuse. Mean-squared curvital and torsional powers are in effect the area enclosed by polar profiles of curvital and torsional power, respectively. The full character of dioptric power cannot be represented by a single vector in the usual sense of the term. Two vectors are required: they are the meridional (vector) power and the orthogonal (vector) power, both of which are associated with the reference meridian. The power along a meridian (often thought of as a scalar or as two scalars) is a vector, the meridional power. This meridional power has components along (the meridional component of the meridional power) and perpendicular to (the orthogonal component of the meridional power) the meridian. In the literature, these components are the curvital power and the negative of the torsional power, respectively. The paper also examines the generalization of these results to the dioptric power of thick systems. Dioptric power is not a fundamental optical property but a derived property. Divergence, the negative of dioptric power, is the corresponding fundamental property. The theory described here is ray-based. The concept of the wavefront is unnecessary. The many formulas and concepts that apply in the context of dioptric power apply directly to the fundamental properties as well. The theory has the potential to provide a complete framework for future studies of astigmatic systems and could systematize the approach to and enhance the knowledge of astigmatism.

A sample from a population of dioptric powers may be used to estimate the distribution of dioptric powers in the population itself. This paper describes the method and shows further how one can obtain a graphical representation of the distribution. The graphical representation takes the form of ellipsoids of constant probability density. The centroid of each ellipsoid estimates the mean of the population while the size, shape and orientation show the extent and nature of the spread of the population. For illustrative purposes the theory is applied to measurements of refractive status before and after radial keratotomy. The ellipsoids are presented as stereo-pairs. They are useful for comparative and predictive purposes. Thus the ellipsoid that contains 95% of the population after surgery defines the set of refractive errors within which the refractive error of a particular eye can be expected to fall with a probability of 95%.

Block neutralization data since 1949 for the foot-and-mouth disease virus system have been analyzed in terms of a unified mass-action theory for computing the amounts of infectious complexes. Proof that infectious complexes contribute considerably to the assays was obtained by demonstrating a reduction in titer after an additional reaction with anti-Ig antibody before the assay. In the suckling-mice assay with intraperitoneal inoculation, both the data of others and our own on several types indicate that for IgG probably three of the unknown total number of critical sites on the virion must be available for infectivity and death. For IgM, just one of an unknown different total number of critical sites on the virion must be available. In tissue culture infectivity assays the minimum number is two or three, whereas in the bovine tongue assay it could be one or two, but probably two. The difficulties in establishing the at present unknown total numbers of neutralization sites to both IgG and IgM are considerable. However, by the simplest interpretation of the data, the number is estimated to be between 5 and 10 for IgG and perhaps just 1 for IgM. A speculation, consistent with the known virion architecture, is that just 1 of the 12 vertices is uniquely involved in infectivity and death, at least in the suckling-mice assay.