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 AR be a double convex lens, whose axis is q1 O, O2, (see Fig. 1), O, and O, being respectively the centers of its spherical surfaces,

1

Let QR, be a ray of a pencil incident parallel to the axis, and let it be refracted at R, in the direction R, R2 q meeting the axis in 91.

2

2

Let IR1T, IR, T, be tangents at R, and R respectively, draw RM, RM and Im, perpendicular to the axis;

and put RM1 = y1

R2M2 = Y2,

also put the distance R, R2 = t1,

and let μ = refractive index.

Then TR1IRT, represents the virtual prism by which the ray is refracted, and its angle I is thus found:

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

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 Req, we have

2

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 R2 R, 92, as in Fig. 2, the contiguous surfaces of the lenses having the same radius, and q1, R2, M2, O2 being the same points as in Fig. 1, we shall have R2 q the direction of the ray, when it meets the concave lens at R2. Also let

be the direction after refraction at R2, (see Fig. 2.)

emergence at R3,

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

Now when the dispersion by one of these virtual prisms is equal and opposite to that of the other, we shall have

♪ (D-D') = &D - § 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 R1, R2, q1, M1, M2, represent the same points as in Fig. 1;

Calling AB the thickness of the lens at its edge the central thickness

Now if it were necessary to retain the variable quantities y, and y in this expression of the value of t, and similarly y, and y in that of t, we should, from 8 (D- D) = 0, have the radii r1, 72, 73, to be expressed in terms of constants, and y1, y2, ys, or the surfaces would be surfaces of revolution generated by curves of variable curvature, in place of portions of surfaces of spheres.

But the dispersion near the edge of a lens is always very great in amount compared with that near the center of its aperture; and it becomes proportionally more important that it should be accurately corrected for the edge, and especially as the enlargement of the apertures of our telescopes may depend upon it. In the example I have calculated, further on in the paper, and are considerably smaller

a2

271

and

a2

2r2