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It must have struck most persons conversant with the subject, that the effect of the lenses, in an achromatic combination of two or three lenses in contact, must be sensibly different near their edges, on account of the oblique passage of the rays, from what it is near their centers; and this difference will be the more important, as the area of that part of the surface of the lens, with this unconsidered effect, is so much more than that part of the surface near the center for which the common theorem is accurate, or nearly so.

In the present paper, I have investigated the conditions of achromatism in a double object-glass for a ray passing through it at a distance from the center of its aperture, on the supposition that we may neglect powers, of the small quantities which enter the expressions, above the first, and also their products. It is also necessary to consider the thicknesses of the lenses, as that of their edges, for all parts at which the new correction rises to any important magnitude.

I have arrived, by two different methods, at the same result, which involves the expression obtained by the ordinary mode, together with others depending on the thicknesses of the lenses. The spaces, through which a ray has passed within the lenses, have on the achromatism an effect which is precisely similar to that of the distance of the lenses in achromatic eye-pieces. If the lens have great thickness, a ray of light after an oblique passage through the glass, will meet the second surface at a different angle to what it would have done if that thickness had been small; and hence, if we consider a virtual prism to be formed by the tangent planes to the surfaces of the lens, at the points at which the ray is incident and emergent, the angle of this virtual prism will depend on the thickness of the lens, as well as on the radii of the surfaces and the distance of the point of incidence from the center of its aperture. We may easily conceive, that this variable angle of the virtual prism will need more accurate consideration when we pass beyond the ordinary first approximation.

The new correction, which we thus arrive at, supposing the thicknesses of the lenses in a double object-glass such as might arise in practice, is however not very large in magnitude. But nevertheless, if it is important to correct as accurately as possible the spherical aberration, an error, which amounts to any sensible part of it, must also merit discussion.

Of the two methods before spoken of, I shall here detail the one which appears the more regular process, and only indicate the other, which however has the advantage of being more continuous in the

working out.

Let R. A. R. be a double convex lens, whose axis is q, O, O, (see Fig. 1), O, and O, being respectively the centers of its spherical

surfaces,
and O, R, + r.

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Let QR, be a ray of a pencil incident parallel to the axis, and let it be refracted at R, in the direction R, R, q, meeting the axis in q.

Let IR, T, IR, T. be tangents at R, and R, respectively, draw R. M., R. M. and Im, perpendicular to the axis; and put R, MI = y, R. M. = y, also put the distance R, R = ti,

and let u = refractive index.

Then T. R. I.R.T. represents the virtual prism by which the ray is refracted, and its angle I is thus found:

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since these angles are always small. Vol. VI. PART III. 4 B

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since the incident rays are parallel, and neglecting the aberration, &c.:

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and since the angle of incidence and the angle of the prism are always small, if D = the angular deviation of the ray, after emergence in the direction R, q, we have

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Again, if we consider the last lens as the outer lens of a double achromatic object-glass, and that the ray emerging immediately into a double concave lens is again refracted by it in the direction R. R., q., as in Fig. 2, the contiguous surfaces of the lenses having the same radius, and qi, R., M., O, being the same points as in Fig. 1, we shall have R, q, the direction of the ray, when it meets the concave lens at R. Also let

R, R, q, be the direction after refraction at R., (see Fig. 2.)
R. 4, .............................. emergence at R,

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Then, as before, the angle of the virtual prism = z I = - 4 R, O, M. -- z R. O. M.

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and, if D be the angular deviation caused by this virtual prism, we have

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p2 a rip, a' rap, u’r, r. Now when the dispersion by one of these virtual prisms is equal and opposite to that of the other, we shall have 3 (D– D') = & D 3 D = 0. Before however we find the actual value of this expression, we must

shew that we may take t and t, as constant quantities, namely, the thicknesses of the lenses at their edges.

In Fig. 3, let It, R., qi, M., M., represent the same points as in Fig. 1;

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