Search

In the presence of astigmatism a focal point typically becomes the well-known interval of Sturm with its pair of axially-separated orthogonal line singularities. The same is true of nodal points except that the issues are more complicated: a nodal point may become a nodal interval with a pair of nodal line singularities, but they are not generally orthogonal, and it is possible for there to be only one line singularity or even none at all. The effect of astigmatism on principal points is the motivation behind this paper. The three classes of cardinal points are defined in the literature in a disjointed fashion. Here a unified approach is adopted, phrased in terms of rays and linear optics, in which focal, nodal and principal points are defined as particular cases of a large class of special structures. The special structures arising in the presence of astigmatism turn out to be described by mathematical expressions of the same form as those that describe nodal structures. As a consequence everything that holds for nodal points, lines and other structures now extends to all other special points as well, including principal points and the lesser-known anti-principal and anti-nodal points. Thus the paper unifies Gauss's and Listing's concepts of cardinal points within a large class of special structures and generalizes them to allow for refracting elements which may be astigmatic and relatively decentred. A numerical example illustrates the calculation of cardinal structures in a model eye with astigmatic and heterocentric refracting surfaces.

The visual axis of the eye has been defined in terms of nodal points. However, astigmatic systems usually do not have nodal points. The purpose of this note is to offer a modified definition of visual axes that is in terms of nodal rays instead of nodal points and to show how to locate them from knowledge of the structure of the eye. A pair of visual axes (internal and external) is defined for each eye. The visual axes then become well defined in linear optics for eyes whether or not they are astigmatic or have decentred elements. The vectorial angle between the visual axes and the optical axis defines the visuo-optical angle of the eye.

The typical stigmatic optical system has two nodal points: an incident nodal point and an emergent nodal point. A ray through the incident nodal point emerges from the system through the emergent nodal point with its direction unchanged. In the presence of astigmatism nodal points are not possible in most cases. Instead there are structures, called nodes in this paper, of which nodal points are special cases. Because of astigmatism most eyes do not have nodal points a fact with obvious implications for concepts, such as the visual axis, which are based on nodal points. In order to gain insight into the issues this paper develops a general theory of nodes which holds for optical systems in general, including eyes, and makes particular allowance for astigmatism and relative decentration of refracting elements in the system. Key concepts are the incident and emergent nodal characteristics of the optical system. They are represented by 2 x 2 matrices whose eigenstructures define the nature and longitudinal position of the nodes. If a system's nodal characteristic is a scalar matrix then the node is a nodal point. Otherwise there are several possibilities: Firstly, a node may take the form of a single nodal line. Second, a node may consist of two separated nodal lines reminiscent of the familiar interval of Sturm although the nodal lines are not necessarily orthogonal. Third, a node may have no obvious nodal line or point. In the second and third of these classes one can define mid-nodal ellipses. Astigmatic systems exist with nodal points and stigmatic systems exist with no nodal points. The nodal centre may serve as an approximation for a nodal point if the node is not a point. Examples in the Appendix, including a model eye, illustrate the several possibilities.

In the standard refraction routine, a Jackson cross-cylinder is reversed in front of the eye by the process of flipping it about its handle. Reverse telescopes have applications in low vision. In ophthalmoscopy, the practitioner views a patient's retina using light that traverses the eye in a direction opposite to that involved in vision. These represent examples of flipped systems that are involved in optometry. The purpose of this paper is to examine what happens to the optical character of an arbitrary optical system when it is flipped in this manner. The analysis is in terms of the ray transference of the system and, because the transference completely characterises the first-order optics of a system, is complete within the limitations of linear optics. It allows for elements in the system to be astigmatic and decentred. An expression is derived for the transference of the flipped system in terms of the transference of the system itself. Expressions are also obtained for the fundamental first-order optical properties and the dioptric power of the flipped system. Three numerical examples are given in the Appendix.

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.

The ray transference matrix completely characterises the first-order optical nature of an optical system including the eye. It is in terms of the transference that quantitative analyses (for example, calculation of an average eye) can be performed. However, the fact that the entries of the transference do not have the same physical dimensions precludes the calculation of the usual scalar value (a Frobenius norm for example) for a change or difference between two optical systems. The purpose of this note is to show how to use the wavelength of the light as a natural unit of length to define a dimensionless transference and so make it possible to calculate a meaningful norm. In most practical applications, some components of the dimensionless transference may dominate unreasonably in the resulting norm in which case suitably weighted norms may be more appropriate. In an appendix, some of the issues are illustrated by application to a lens.

In quantitative analyses of the optical character of eyes (and related systems) it is sometimes necessary to deal with left and right eyes in the same context. In accounting for anatomical symmetry (mirror symmetry in the mid-sagittal plane) one treats a cylinder axis at 20 degrees , say, in a left eye as equivalent to an axis at 160 degrees in a right eye. But this is only one aspect of the linear optical character of an eye. The purpose of this note is to show how to account for anatomical symmetry in the linear optical character of eyes in general. In particular the note shows how to modify the optical properties of left (or right) eyes so that anatomical symmetry is accounted for in quantitative analyses in contexts in which both left and right eyes are involved.

From differential geometry one obtains an expression for the curvature in any direction at a point on a surface. The general theory is outlined. The theory is then specialised for surfaces that are represented parametrically as height over a transverse plane. The general ellipsoid is treated in detail as a special case. A quadratic equation gives the principal directions at the point and, hence, the principal curvatures associated with them. Equations are obtained for ellipsoids in general that are generalisations of Bennett's equations for sagittal and tangential curvature of ellipsoids of revolution. Equations are also presented for the locations of umbilic points on the ellipsoid.

Considering a set of eyes, how does one define an average whose optical character represents an average of the optical characters of the eyes in the set? The recent proposal of the exponential-mean-log-transference was based on a conjecture. The purpose of this note is to provide justification by proving the conjecture and defining the conditions under which it fails.

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.

A general expression is derived for the proximity factor in near image size magnification for an arbitrary instrument in front of an arbitrary eye. The proximity factor is a 2 x 2 matrix. The instrument and eye may be astigmatic and have decentred elements. The image on the retina may be blurred or not. The analysis is exact within the limitations of linear optics. The general results are specialized for the case of a stigmatic instrument and a stigmatic eye. The results are applied to the case of a thick, possibly bitoric, spectacle lens. The Appendix treats two numerical examples.

Traditional treatments of spectacle magnification for distant objects consider only stigmatic spectacle lenses and they compare the retinal image size in a refractively fully compensated eye with the image size in the uncompensated eye. Spectacle magnification is expressed as a product of two factors, the power and shape factors of the lens. The power factor depends on the position of the entrance pupil of the eye. For an eye with an astigmatic cornea, however, the position of the entrance pupil is not well defined. Thus, the traditional approach to spectacle magnification does not generalize properly to allow for astigmatism. Within the constraints of linear optics and subject to the restriction that the eye's iris remains the aperture stop, this paper provides a complete, unified and exact treatment for optical instruments in general. It compares retinal image size in a generalized sense (including image shape and orientation) for any instrument in front of an eye with that of the eye alone irrespective of whether the instrument compensates or not. The approach does not make use of the concept of the entrance pupil at all and it allows for astigmatism and for non-alignment of refracting elements in the instrument and in the eye. The concept of spectacle magnification generalizes to the concept of instrument size magnification. Instrument size magnification can be expressed as the product of two matrix factors one of which can be interpreted as a power factor (as back-vertex power) and the other factor for which the name dilation factor is more appropriate in general. The general treatment is then applied to a number of special cases including afocal instruments, spectacle lenses (including obliquely crossing thick bitoric lenses), contact lenses, stigmatic systems and stigmatic eyes. In the case of spectacle lenses, the dilation factor reduces to the usual shape factor.

The principal meridians of the powers of thick astigmatic systems, like the eye, are not necessarily at right angles. The consequence is a class of phenomena included in the category commonly described as irregular astigmatism. The conventional principal meridional representation of power, however, is unsuited to quantitative analysis. This paper presents equations for converting from the principal meridional form of power to a representation, the dioptric power matrix, which is amenable to quantitative analysis. It generalizes an earlier paper which treated only powers of a conventional form in which the principal meridians are always at right angles. It copes in particular with what are known as asymmetric powers. A routine is also presented for converting in the reverse direction, from the power matrix to the principal meridional form of power. The principal meridional form of power turns out not always to be unique, there being distinct powers (they are asymmetric) with the same principal powers and meridians. Thus, in general, the dioptric power matrix is a satisfactory representation of power while the principal meridional representation is not.

Converting the traditional representation of power as sphere, cylinder and axis to the dioptric power matrix F is usually performed by means of Long's equations and the reverse process by means of Keating's equations. It is sometimes useful to be able to convert directly between the matrix and power expressed in terms of principal powers F1 and F2 along corresponding principal meridians at angles a1 and a2. The equations for interconverting F and the principal-meridional representation expressed as F1(a1)F2 are presented here. Equivalent equations allow direct interconversion of the reduced vergence matrix L and the principal-meridional representation of vergence L1(a1)L2. Vergence becomes infinite at line and point focuses. Similarly effective power and back- and front-vertex power are infinite for some systems. Nevertheless it is possible unambiguously to represent infinite vergence and vertex power in principal-meridional form. However, information is usually lost in these infinite cases when the principal-meridional representation is converted to the matrix representation, and the former is not recoverable from the latter. As a consequence the matrix representation is usually unsatisfactory for vergences and vertex powers that are infinite. On the other hand, the principal-meridional representation of vergence and power is always satisfactory. If one adopts the position that effective powers and vertex powers are really vergences rather than powers then one concludes that the matrix provides a satisfactory representation for powers of thin systems in general but not for vergences. Implied by a vergence at a point is an interval of Sturm. The equations for characterizing the interval from the reduced vergence are presented.

The traditional step-along vergence procedure applies to stigmatic systems, that is, systems that are not astigmatic. Computation is disrupted when a focus coincides with a thin lens or refracting surface. A small change to the procedure results in a modified procedure which overcomes the computational problems. The modified procedure is easier to execute than the traditional procedure and allows one to write down useful equations directly. Among the formulae are those for back-vertex power. A step-along vergence procedure also exists for astigmatic systems. It makes use of the dioptric power matrix and the reduced vergence matrix. Computational problems arise when a point or line focus coincides with a thin lens or refracting surface; however they are not overcome by an analogous modification to the procedure. Nevertheless the modified procedure has some advantages including the fact that, as for stigmatic systems, it allows one to write down useful formulae directly. Stepwise calculations of vergence are sometimes performed backward through a system; the advantages and disadvantages described for step-along procedures holds for such step-back procedures as well.

The quantitative representation and analysis of astigmatism present difficulties for the researcher. The case is made that the difficulties arise because of the way astigmatism is conceived and defined. In most cases astigmatism is regarded as cylinder. Cylinder, however, is not invariant under spherocylindrical transposition and, hence, cannot strictly be regarded as meaningful. The purpose of the paper is to find a rational, context-free, invariant and universally-applicable definition of astigmatism for all quantitative analyses. One is led to definitions of astigmatism and its components some of which have already appeared in the literature but which are not in use in analyses of astigmatism. Astigmatism is defined with respect to pure sphere. In the case of thin systems (including keratometric measurements and refraction) astigmatism turns out to be Jacksonian power, that is, the power of a Jackson crossed cylinder. The power of every thin system can be regarded as consisting of two orthogonal components, sphere and astigmatism. The astigmatism of thin systems itself further decomposes naturally into two orthogonal components called ortho- and oblique astigmatism. In the case of thick systems, like the eye itself, astigmatism decomposes naturally into three orthogonal components, ortho-, oblique and antisymmetric astigmatism. The approach is based on the general definition of power in paraxial optics, the dioptric power matrix, and leads to useful graphical representations. Because of its mathematical foundation the analysis can claim completeness and contextual independence. Furthermore it is also directly applicable to the four fundamental paraxial properties of optical systems.

Variation of keratometric measurements on an approximately spherical surface on a polymethyl methacrylate (PMMA) button is compared to keratometric variation on the PMMA button after the application of an artificial solution. The variation is shown by means of scatter plots in a three-dimensional space, as well as quantitatively in the form of variance-covariance matrices. The wet button showed variances a factor of more than 1000 times that of the dry button. This suggests that the precorneal tear film could be an important, often-overlooked contributing factor to keratometric variation in the case of the eye.

The problem of locating the points of maximum and minimum thickness at the edge of a lens, and of calculating the thickness at those points, is examined for lens powers and for edge shapes in general. The edge extremal problem, as the problem is called, is solved explicitly for general powers along straight cut edges. The extremal problem is also analysed for lenses with circular and elliptical edges but explicit solutions are obtained only for centrally cut lenses, that is, lenses with coincident optical and geometrical centres. For edges that are neither straight nor centrally cut ellipses (including circles), it appears that explicit solutions cannot be obtained for the edge extrema: one has to resort to numerical solution of implicit equations or to the calculation of thickness at sufficiently many points around the edge. For the centrally cut ellipse the edge extremal problem turns out to be the eigenvalue problem of linear algebra. In general the thickness extrema at the edge do not lie on the principal meridians of the lens, nor do they lie on meridians that are mutually perpendicular. With minor modification the results apply equally well to the edge extrema for sagitta of a surface.

A matrix expression is derived for the prismatic effect in the reading portion of a bifocal lens. Using this expression the problem of finding the location of the optical centre of the reading portion is examined. Matrix methods lead to a complete solution that holds in all circumstances within the usual limitations of first-order optics and thin lenses. Both the main and segment lenses may be astigmatic. In the usual case there is a unique near optical centre but situations can occur in which there is no near optical centre or an infinite number of them. These situations are examined systematically. Numerical examples illustrate the method.

Ocular movements required in most visual functions are affected by lenses in front of the eye. Fundamental to an understanding of these movements are concepts such as the ocular rotation factor and the ocular orientational demand. In this paper the ocular rotation factor for thin spherical lenses is generalized for astigmatic lenses. A compact matrix equation is obtained for the orientational demand in the case of such lenses. The equations are satisfactory in ordinary situations but they break down under certain circumstances. Equations are derived for these special circumstances in particular and also for all circumstances in general.

Meridional profiles of variation of dioptric power are constructed. Using the basic theory developed in an accompanying paper, samples are selected in a systematic way to illustrate variation only in sphere, only in cylinder and only in axis, and in all possible combinations of sphere, cylinder and axis. For each of the seven samples, scatter plots are constructed together with ellipsoids that represent the estimated distribution of powers in the population from which the sample was taken. The surfaces of the ellipsoids are surfaces of constant probability density within which 95% of the population is calculated to lie. The scatter plots and distribution ellipsoids are plotted in a three-dimensional space called h-space. Meridional profiles of variation are constructed for each of the samples. Properties of the profiles are discussed. Meridional profiles are also presented for eyes before and after radial keratotomy. Among other things, the profiles show meridians of greatest and least variation and are intuitively satisfying. They are potentially useful for the researcher and clinician including the surgeon. They may help to improve surgical and therapeutic techniques. Certain patterns may prove to be characteristic of physiological or pathological conditions, in which case meridional profiles may have use as a diagnostic tool.

It has recently become possible to calculate and represent variation or spread of dioptric power in a meaningful way. This is important for the proper analysis and interpretation of data on dioptric power in a number of areas of the vision sciences. The representation takes the form of a symmetric matrix of six (usually) variances and covariances. Although the matrix is satisfactory for several formal statistical purposes, such as the testing of hypotheses, it does not give an intuitively satisfactory picture of the extent and nature of the variation, nor is it easy to interpret in a way that could be useful to the researcher or clinician. A useful graphical representation of variation can be constructed from the variance-covariance matrix. It consists of curves that show the meridional dependence of the variation. These meridional profiles of variation give a complete and intuitively satisfactory picture of the nature and extent of the variation of power and are potentially of general use to researcher and clinician. A complete theoretical basis is provided for the construction of meridional profiles of variation of dioptric power. An accompanying paper employs the theory to construct profiles for a number of representative samples of dioptric power.

Recently developed methods of quantifying changing dioptric power are applied to the refractive status of Hong Kong Chinese infants from about 10 to about 40 weeks of age. The analysis confirms that hyperopia decreases during the period and that with-the-rule astigmatism predominates. Change, however, is not constant over the period. The rate of decrease of hyperopia slows down and the refractive status becomes more spherical. The mean refractive status is calculated at 10, 20, 30 and 40 weeks. The spread of refractive status is represented by variance-covariance matrices calculated for each group and 95% confidence ellipsoids on the mean are constructed. The variance-covariance exhibits little change over the period. The right and left eyes show similar behaviour.

A recent paper analysed change in refraction and corneal curvature associated with contact lens wear in patients who had had radial keratotomy at least a year beforehand. Conventional methods of analysis were used. This paper applies methods that have only recently become available. The results are clearer and less ambiguous. Formally, the analysis shows that there are significant mean changes in refraction (estimated to be +1.69/-0.50 x 105) and corneal power (-0.66/-0.25 x 155). In spite of the difference between these two mean changes, the analysis shows that the change in refraction may be associated with change in the corneal power alone. There is no reason to believe that any other change is occurring in the eye. Care needs to be executed in assigning a causative role to contact lenses in changing refraction and corneal curvature. Almost certainly the lenses do have such a role but the analysis does not formally allow an unequivocal conclusion concerning that role.

A dioptric power or surface curvature may be presumed or known to be of some particular type, for example spherical, or astigmatic and with a particular nearest equivalent sphere. Such knowledge or presumption represents a constraint on the power or curvature. This paper shows how meridional measurements can be used to obtain least-squares estimates of surface curvature or dioptric power under the constraint.

The calculation or estimation of dioptric power or surface curvature from meridional measurements is an important problem with applications in a number of areas. By phrasing the problem in terms of matrices and applying new results in the mathematics of matrices one is able to reformulate the problem into a standard form that is well known in mathematics and statistics. As a result the solution can be written down directly. The method copes with any number of measurements along any number of meridians, including repeated measurements along the same meridian, and is more general than methods previously proposed. It gives least-squares and best estimates of the true surface curvature or dioptric power. Being phrased in standard statistical terms the method lends itself readily to extension to related types of problems such as least-squares estimation under certain types of constraints. Matrix methods employed in this paper are likely to find wider application in optometry and the vision sciences.

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%.

Methods have been developed recently for testing hypotheses on mean dioptric power and for constructing confidence regions in situations that are most likely to be encountered. In this paper the methods are extended to make the analysis complete. A new situation covered specifically is that of dioptric power not of the form sphere/cylinder x axis. Such powers, termed asymmetric powers because the dioptric power matrices are asymmetric, include the equivalent power of a thick obliquely crossed bitoric lens. A second situation is that in which the covariance matrix of the sample of powers is singular. Symmetric dioptric power (the more familiar form of power) can be represented by a point in three-dimensional space. In general, however, dioptric power is four dimensional in character. Singularity of covariance arises when variation in the sample is limited to a subspace of dimension less than the full three or four. The space spanned by the sample is called the range space of the sample. The dimension of the range space may be four, three, two, one or zero. Each case is considered in turn. Numerical examples of hypothesis testing are presented in range spaces of dimension four to one. The test statistic devised for each case also gives the equation of the confidence region about the mean of a sample of dioptric powers. Singularity can sometimes be avoided merely by taking larger samples and by taking more accurate readings. The problem of near singularity is briefly discussed. The paper allows basic hypothesis testing on mean dioptric power and the construction of confidence regions in all possible circumstances.

A single approximate matrix expression is derived that gives the thickness at any point on any lens. The lens may have dioptric power or prism power or both. The expression gives, amongst other things, the equation for lines of constant thickness.

Every dioptric power of the usual form sphere/cylinder x axis may be represented by means of a point in three-dimensional space. Graphical representation of data in this manner is important for statistical analysis. In particular, graphical representation may be used to display confidence regions about a mean power, for example. A disadvantage of these representations, however, is that simply changing the reference meridian for cylinder axes changes distances in the space and, therefore, changes the shapes of the confidence regions. Because the shapes define the nature of the variation and give, in particular, the principal components of variation, a researcher who happens to measure the orientation of cylinder axes from the 20 degree meridian, for example, instead of the conventional horizontal meridian, could be led to different statistical conclusions. The implication is that such conclusions are unlikely to have much physical meaning. A representation is described here in which distances and shapes do not depend on the meridian that happens to be chosen as reference. Each dioptric power is represented by a point in Euclidean 3-space. Several examples of graphical representation are given. The spherical powers occupy a particular line in the space, the Jackson crossed cylinders occupy a plane, the cylindrical powers occupy a cone, and so on, for all types of conventional dioptric power. These lines and surfaces are illustrated. The statistical implications are discussed briefly. The representation satisfies the requirements of the statistics and is proposed as the standard one for future use.

In general thickness varies along the cut edge of a lens. There is some interest in being able to predict the locations of the thickest and thinnest points on the edge. One purpose of this paper was to formulate the general problem mathematically of finding the locations of thickness extrema on the edge of an arbitrary lens cut into an arbitrary shape. A second purpose was to illustrate how the problem can be solved. In particular, the problem is solved completely and explicitly for what is probably the simplest case, the straight cut edge. A component-free matrix expression for the position of the extremum is derived by employing Lagrange multipliers and the concept of the generalized inverse of a matrix. The equation applies to spheres, cylinders and sphero-cylinders and allows for the presence of prism as well. Along some edges the thickness is constant. Along other edges the thickness varies linearly; there are no extrema except for the practical extrema at the two ends of the edge: thickest at the one and thinnest at the other. The mathematical conditions for these two cases are presented. Equivalent to the matrix equation for the position vector of the thickness extremum is a pair of scalar equations expressed in terms of components of the matrices. The pair is useful when locating extrema using manual calculations. On the other hand, the component-free matrix equation is useful in other circumstances such as for mathematical manipulations and when using computer software that handles matrices. The former is used in a number of numerical examples; the latter was used to check the answers by computer. The mathematical techniques described here are likely to find application in other areas of ophthalmic and visual optics.

It has not hitherto been possible to apply formal methods of statistical analysis to data on dioptric powers. The solution to the basic statistical problem is now provided in this paper. Recognition of the matric-variate nature of dioptric power allows calculation of sample means and variance-covariances. These in turn can be used to calculate a statistic for testing hypotheses on population means and for obtaining confidence regions for those means. In a graphical representation of dioptric power the confidence region turns out to be an ellipsoid centred on the mean of the sample of dioptric powers. The theory is illustrated by means of numerical examples. Singularity of the variance-covariance matrix may occur especially when the sample is small. When it does occur it is the cause of some difficulty in applying the statistics. Nevertheless singularity is rare in practical situations and can usually be avoided simply by increasing the size of the sample. Singularity, therefore, is not treated fully in this paper. Dioptric power is essentially four-dimensional in character but in practice a three-dimensional subspace is almost always sufficient. To avoid the difficulty of having to represent four-dimensional shapes and to avoid the complication of singularity (which is the rule rather than the exception in practice in four-space) only the common three-dimensional problem is considered in detail.

The important formal process of inferring properties of populations from statistics of samples requires a method by which variables can be compared or ranked. A suitable method of comparing dioptric powers is described. The method recognizes the essentially four-dimensional nature of dioptric power and is sufficiently general to allow for comparison of powers (for example, equivalent powers of general thick bitorics) that cannot be represented in the usual manner, sphere/cylinder x axis.

Matrices can be multiplied in several ways. As a result one can define a number of distinct squares of the dioptric power matrix. Additional squares can be defined for matrices vectorized by means of the vec and vech operators. These various squares can form the basis for the definition of variance and covariance of samples of dioptric powers. The complete form of the variance of dioptric powers is a 4 x 4-matrix (a variance-covariance matrix) with 10 distinct elements; four of them represent the variances of the four elements of the dioptric power matrix and the other six are the covariances between those four elements. For thin systems there are only six distinct elements of the variance-covariance matrix, three of which are variances and three covariances. Examples are included that show the calculation of the different types of squares and the variance-covariance matrix for a sample of equivalent powers of thick bitoric lenses and for a sample of powers of thin systems, including conventional refractive errors. Variances calculated in the past, including those of nearest equivalent spheres and Gartner's and Churm's variances turn out to be components (or combinations of components) of the generalized variance defined here. They are valid as far as they go but they do not completely represent the dispersion of a sample of dioptric powers. The complete variance-covariance matrix does represent the dispersion fully and thus opens the way for the formal statistical analysis of measurements of dioptric power.

The dioptric power matrix of a single refracting surface is shown to be directly related to the differential form known in differential geometry as the second fundamental form. Consequently it is a complete description of the local refractive character of the surface. Generalized equations are derived for sagitta and lens thickness for surfaces of any form. The equations hold approximately for points close to the optical axis except in the case of paraboloidal surfaces: for paraboloidal surfaces they hold exactly. Lens thickness approximately obeys what is termed the principle of form independence. The value of the second fundamental form is approximately double the sagitta. Formal support is provided for Keating's concept of a generalized matric vergence of astigmatic wave-fronts.