compels the eyes to diverge in order to obtain binocular vision of an object placed at a distance. Such an instrument, the principle of which has already been pointed out by Herschell, can be obtained by superposing too prisms of equal strength, which can be rotated in opposite directions. It may be purchased, mounted on a handle, on which is given the degrees of the prism resulting from the union of the two component glasses. To the above scale, I have added two others, on the rim of the instrument, which give, for a base-line of 64 millimetres, and for one of 58 millimetres, the number of metre-angles necessary for each eye to overcome a prism of given strength. It is not difficult to show the relation existing between the strength of any prism and the number of metre-angles which expresses the deviation produced. For a base-line (or distance between the centres of rotation of the two eyes) of 58 millimetres, as in children, a metreangle corresponds to 1° 39′ 39′′-say 100'. The deviation produced by a prism may be taken as equal to half its angle of opening, which is marked on each prism in our trial cases, or on the hand of the double prism. Therefore a prism of X° will produce a deviation of or of X° X x 60' It is necessary only to divide this value by 100, in order to obtain the corresponding number of metre-angles: X x 60' 3 X This formula, reduced to its simplest expression, becomes that is to say, we have only to multiply the number of the prism by 3, and divide the prism by 10, in order to find, in metre-angles, the deviation for a base-line of 58 millimetres. When the prism is held before one eye only, as in the determination of the minimum of conveyance by the double prism, its action is divided between the two eyes. The total deviation 3 X each eye. A prism of 6° produces a deviation of 18 20 = 10 3 X gives for angles. But, if both eyes concur to neutralise its effect, each eye need only change its direction 0.9 metre-angle. It is only when a prism of 6° is placed before each eye that the full result of 18 metreangles is obtained; always, of course, for a base-line of 58 millimetres. When the base-line is longer, for example 64 millimetres, as in adults, the metre-angle becomes 1° 50' 110', and the formula becomes 3 for the deviation corresponding to the prism of X°, or effect produced on each eye, when the prism is placed before one only. X 3 X The mean between these two fractions, and 20 3 X for the In other words, we may simply divide the number of the prism by 7, in order to obtain approximately, in metre-angles, the amount of deviation of each eye, when the prism is placed before one. Two prisms of equal power, one before each eye, necessarily produce a double effect, or 2 X When, therefore, the punctum remotum of convergence is situated at an infinite distance or beyond, the minimum of convergence is determined by the following method. Making the patient fix the flame of a candle at a distance of 6 metres or more, we find the strongest abducting prism, which, placed before one eye, can be overcome without causing homonymous diplopia. We then read off, on my scale attached to the double prism, the minimum of convergence in metre-angles, or we deduce it from the number of the prism, by means of the method just described. Here, too, it may often be necessary to have recourse to a coloured glass, in order to render the patient attentive to his diplopia. If the maximum (p) and the minimum (r) be found, the amplitude of convergence (a) is obtained, as we have already explained, by subtracting the latter from the former: a=p-r. IV. DETERMINATION OF THE RELATION BETWEEN THE CONVERGENCE AND THE OPTICAL ADAPTATION IN BINOCULAR VISION. Our dynamometer also serves to ascertain whether or not proper binocular vision exists for a definite distance. For this purpose its funnel bears a series of small holes arranged in a vertical line. As long as the convergence and optical adaptation of the eyes are correct (with or without glasses), the holes are seen single and distinctly. If the refraction does not correspond to the distance of the object, the points appear indistinct, but the line formed by them still remains single as long as the eyes are normally directed. If the convergence is insufficient, but the optical adaptation correct, the line appears double, while the points are distinct. If, however, both functions are at fault, there will be, at the same time, both diplopia and indistinctness of vision. The line appears double, and its points are blurred, and form discs of diffusion. V. DETERMINATION OF THE RELATIVE RANGE OF ACCOMMODATION AND CONVERGENCE. We have explained, p. 196 et seq., what is meant by the relative range of accommodation and convergence. The problem of their determination is as follows: 1st. The convergence remaining unaltered, by how many dioptries can the accommodation be increased or diminished? 2d. The accommodation remaining unaltered, by how many metreangles can the convergence be increased or diminished? These determinations were first undertaken by Donders and MacGillavry, by means of an apparatus consisting of a quadrangular board, 162 centimetres long and 24 centimetres broad, kept in a horizontal position by a standard capable of being lowered or raised at will. One end of the board has a notch for the nose of the person under examination. The position of the eyes is determined by two wooden rods, rounded at the ends and drawn out at pleasure, against which the cheeks rest. From in front of each eye a divided groove extends the whole length of the board, perpendicular to the base-line. In front of each eye is a grooved half-circle for holding lenses, and moveable in an arched groove, whose centre of curvature is situated on an imaginary prolongation of the horizontal groove, within the eye. The eyes should be so placed that their nodal points may, as nearly as possible, be over the line passing through the zero-points of the divisions of the longitudinal groove, and so that their centres of rotation may be considered as coincident with the centre of the abovementioned arcs. Then the lenses may always be given such a position that their axis will correspond to the lines of fixation. Two small microscopes, attached to the board, indicate, by the crossing-points of their micrometric threads, the position of the corneæ of the eyes under examination, and are useful in watching and controlling the position of the latter. The half-rings, which contain the lenses, can be moved toward, or away from, each other, so that the distance between the lenses may always correspond to the distance between the eyes of the person under examination. From the middle of the space between the eyes, a third longitudinal groove runs parallelly with the two above mentioned. In it slides a piece of wood destined to carry the object of fixation. The latter consists of a few hairs stretched across a small frame, or of some very small holes in a metallic diaphragm. Distinct vision of this object demands a greater or less effort of convergence, in proportion as the object is nearer to, or further from, the eyes. In apparatus of this kind, as found in trade, the median groove is marked with two scales, indicating the angles of convergence for two different distances between the centres of rotation,-—that is, for two base-lines, one 64 and the other 54 millimetres in length. 1 Donders, Holländische Beiträge, &c., vol. i., p. 379, 1846, and MacGillavry, Onderzoekingen over de hægrootheid der Accommodatie: Inaug. diss. Utrecht, 1858. Donders' Anomalies, &c., p. 97. T To determine the maximum of relative convergence (p) we place before the eyes concave glasses of successively increasing strength, which require an increasing effort of accommodation. To find the minimum of relative convergence (r1), convex glasses, on the contrary, are used. The strongest concave glass that can be borne, without prejudice to binocular and distinct vision, gives the positive portion, and the strongest convex glasses the negative portion of the amplitude of relative accommodation; and the difference between them constitutes the relative amplitude of accommodation, a,. These experiments are made for different degrees of convergence, and the results noted on diagrams,-i.e., on a system of co-ordinates. On the horizontal lines (abscissa) are recorded the degrees of convergence; on the vertical lines (ordinata), the degrees of accommodation. Hence the figures placed on the vertical line, at the left of the diagram, indicate metre-angles; and those at the bottom, on the horizontal line, dioptries. We have already given and explained a series of such diagrams (pp. 202-218, Figs. 82-88). CHAPTER IV. ASTIGMATISM. I. REGULAR ASTIGMATISM. HERETOFORE We have considered the refractive surfaces of the eye as surfaces of revolution,-i.e., as produced by the rotation of an ellipse, or circle, about the optic axis. A surface of this kind has, necessarily, the same curvature in all its meridians; every plane passed through the axis is limited by the same curve, viz., the curve which, by its rotation, has engendered the surface. From this it follows that light is equally refracted in each of these meridians. Rays emanating from a luminous point are focused on the axis by each meridian, and at a single point of the image. The union of the foci of all the meridians constitutes the focus of the surface. It is true that we made a certain restriction in the case of a spherical surface (p. 12). We said that the farther the zone, through which the rays pass, is from the axis, the nearer to the surface their focus will be; so that rays passing through a sphere are not all united exactly at one point. But they are, at least, all focused on the axis. The focal line formed by them, in the least favourable case, coincides with this optic axis. Moreover, if the image, of a luminous point, formed by a spherical surface, be received upon a screen held perpendicularly to the axis, this image is everywhere a circle, or a larger or smaller luminous point,-i.e., always similar to its object. The focusing is more nearly perfect for an ellipsoid of revolution than for a sphere, since, on account of the gradual diminution in curvature of the ellipsoid, the eccentric rays are directed toward the same point as those passing nearer the axis. The refractive surfaces of a perfectly formed eye are very like an ellipsoid of revolution, called, in this case, ellipsoid with two axes. One of them, the major axis of the ellipse, is, at the same time, the optic axis and that of rotation; the other is perpendicular to it and is equal in all meridians. When speaking of the form of the dioptric surfaces of the eye, we said that eyes so perfectly constructed are rarely found; that the curvature of the cornea, for instance, is nearly always greater in some |