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[From the Report of the British Association for 1849. Part II. p. 10.]

ON A MODE OF MEASURING THE ASTIGMATISM OF A DEFECTIVE EYE.

BESIDES the common defects of long sight and short sight, there exists a defect, not very uncommon, which consists in the eye's refracting the rays of light with different power in different planes, so that the eye, regarded as an optical instrument, is not symmetrical about its axis. This defect was first noticed by the present Astronomer Royal in a paper published about 20 years ago in the Transactions of the Cambridge Philosophical Society. It may be detected by making a small pin-hole in a card, which is to be moved from close to the eye to arm's length, the eye meanwhile being directed to the sky, or any bright object of sufficient size. With ordinary eyes the indistinct image of the hole remains circular at all distances; but to an eye having this peculiar defect it becomes elongated, and, when the card is at a certain distance, passes into a straight line. On further removing the card, the image becomes elongated in a perpendicular direction, and finally, if the eye be not too long-sighted, passes into a straight line perpendicular to the former. Mr Airy has corrected the defect in his own case by means of a spherico-cylindrical lens, in which the required curvature of the cylindrical surface was calculated by means of the distances of the card from the eye when the two focal lines were formed. Others however have found a difficulty in preventing the eye from altering its state of adaptation during the measurement of the distances. The author has constructed an instrument for determining the nature of the required lens, which is based on the following proposition:

Conceive a lens ground with two cylindrical surfaces of equal radius, one concave and the other convex, with their axes crossed

at right angles; call such a lens an astigmatic lens; let the reciprocal of its focal length in one of the principal planes be called its power, and a line parallel to the axis of the convex surface its astigmatic axis. Then if two thin astigmatic lenses be combined with their astigmatic axes inclined at any angle, they will be equivalent to a third astigmatic lens, determined by the following construction :-In a plane perpendicular to the common axis of the lenses, or axis of vision, draw through any point two straight lines, representing in magnitude the powers of the respective lenses, and inclined to a fixed line drawn arbitrarily in a direction perpendicular to the axis of vision at angles equal to twice the inclinations of their astigmatic axes, and complete the parallelogram. Then the two lenses will be equivalent to a single astigmatic lens, represented by the diagonal of the parallelogram in the same way in which the single lenses are represented by the sides. A plano-cylindrical or spherico-cylindrical lens is equivalent to a common lens, the power of which is equal to the semisum of the reciprocals of the focal lengths in the two principal planes, combined with an astigmatic lens, the power of which is equal to their semi-difference.

If two plano-cylindrical lenses of equal radius, one concave and the other convex, be fixed, one in the lid and the other in the body of a small round wooden box, with a hole in the top and bottom, so as to be as nearly as possible in contact, the lenses will neutralize each other when the axes of the surfaces are parallel; and, by merely turning the lid round, an astigmatic lens may be formed of a power varying continuously from zero to twice the astigmatic power of either lens. When a person who has the defect in question has turned the lid till the power suits his eye, an extremely simple numerical calculation, the data for which are furnished by the chord of double the angle through which the lid has been turned, enables him to calculate the curvature of the cylindrical surface of a lens for a pair of spectacles which will correct the defect of his eye.

[The proposition here employed is easily demonstrated by a method founded on the notions of the theory of undulations, though of course, depending as it does simply on the laws of reflection and refraction, it does not involve the adoption of any particular theory of light.

Consider a thin lens bounded by cylindrical surfaces, the axes of the cylinders being crossed at right angles. Refer points in the neighbourhood of the lens to the rectangular axes of x, y, z, the axis of z being the axis of the lens, and those of x and y parallel to the axes of the two cylindrical surfaces respectively, the origin being in or near the lens, suppose in its middle point. Let r, s, measured positive when the surfaces are convex, be the radii of curvature in the planes of xz, yz respectively. Then if T be the central thickness of the lens, the thickness near the point (x, y) will be

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very nearly. As T'is constant, and is supposed very small, we may neglect it, and regard the thickness as negative, and expressed by the second term in the above formula. The incident pencil being supposed to be direct, or only slightly oblique, and likewise slender, the retardation of the ray which passes through the point (x, y) may be calculated as if it were incident perpendicularly on a parallel plate of thickness

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so that if R be the retardation, measured by equivalent space in air, and μ be the index of refraction

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( + - ) (x2 + 3′) + (μ − 1) . 1 (¦— — —1) (∞° — y2).

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y

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The effect therefore of our lens, to the lowest order of approxi mation, which gives the geometrical foci in the principal planes, is the same as that of two thin lenses placed in contact, one an ordinary lens, and the other an astigmatic lens. If r' be the radius of curvature of the plano-spherical lens equivalent to the ordinary lens, and " that of the astigmatic lens, we have

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as above enunciated. If p be the power of the astigmatic lens,

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If two thin astigmatic lenses of powers p, p' and with their astigmatic axes inclined at azimuths a, a to the axis of y be combined, we shall have for the combination

- R = 1 pp2 cos 2 (0 − a) + 1⁄2 p'p2 cos 2 (0 — a'),

which is the same as would be given by a single astigmatic lens of power p, at an azimuth a,, provided

pp2 cos 2 (0-a) + p'p2 cos 2 (0-a') = p,p2 cos 2 (0 - a1),

which will be satisfied for all values of 0 provided

p cos 2a+p' cos 2x′ =p1 cos 21⁄4 ̧,

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These two equations geometrically interpreted give the proposition enunciated above for the combination of astigmatic lenses.]

[From the Report of the British Association for 1849. Part II. p. 11.]

ON THE DETERMINATION OF THE WAVE LENGTH CORRESPONDING WITH ANY POINT OF THE SPECTRUM.

MR STOKES said it was well known to all engaged in optical researches that Fraunhofer had most accurately measured the wave lengths of seven of the principal fixed lines of the spectrum. Now he found that by a very simple species of interpolation, which he described, he could find the wave length for any point intermediate between the two of them. He then exemplified the accuracy to be obtained by his method by applying it to the actually known points, and shewed that in these far larger intervals than he ever required to apply the method to the error was only in the eighth, and in one case in the seventh, place of decimals. By introducing a term depending on the square into the interpolation still greater accuracy was attainable. The mode of interpolation depended on the known fact that, if substances of extremely high refractive power be excepted, the increment Au of the refractive index in passing from one point of the spectrum to another is nearly proportional to the increment Aλ of the squared reciprocal of the wave length. Even in the case of flint glass, the substance usually employed in the prismatic analysis of light, this law is nearly true for the whole spectrum, and will be all but exact if restricted to the interval between two consecutive fixed lines. Hence we have only to consider μ as a function, not of λ, but of λ2, and then take proportional parts.

On examining in this way Fraunhofer's indices for flint glass, it appeared that the wave length Bλ of the fixed line B was too great by about 4 in the last, or eighth, place of decimals. It is

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