with 7 for a ray which passes near the edges, and are therefore negligible; and also is again a much smaller quantity, and 1.2. (q. M1) therefore not needing attention: so that we shall consider t, and t as the thicknesses of the lenses at or near their edges, and as constants.

Differentiating the expressions for D and D', and substituting in the expression & (D - D') D') = 0, we find, after the reductions,

In the ordinary formula, the condition that a double object-glass shall be achromatic, is

So that the expression we have obtained consists of the common one together with other terms involving the thicknesses of the lenses.

The other method which I have mentioned will be easily comprehended from Fig. 4.

If abed in this figure represent a double object-glass, and R1, R2, R3, M1, M2, M3, 91, 92, 93, represent the same points as in the two first figures; R R2 Rq3 being the path of the ray incident at R1, then we have, tangent of the angle R, q M ̧, in which the emergent ray meets the axis at 93,

2

and, in order that the combination may be achromatic, we must have the variation of this R3q3 M3 (or its tangent since it is small) = 0, whilst μ and ', the refractive indices, vary for the different colours of the spectrum. For this purpose, expressing ys in terms of y, and the radii of the surfaces, and distances RR2, RR, in the lenses; and qM3

in terms of the radii and distances; and then performing the required differentiation, we find, for d.tan R3q3 M3, the same expression as we have just obtained for (D-D'), as we clearly ought to do.

To enable us to judge of the value of the new correction, it is necessary to apply it to a case which may arise in practice. For this purpose I have chosen the third case in Sir John Herschel's table in the paper before referred to; as the dispersive ratio in that case is what he considers the mean value for such glass as is usually obtained in England. I have also considered the radii of the interior surfaces to be the same, their difference being little more than a fiftieth of an inch in three feet, so that we have for our data for a telescope of ten feet focal length, as follows:

If we take, now, the same dimensions, for our example, as those of the Northumberland telescope recently put up at the Cambridge Observatory, in which the focal length is 19 feet, the aperture 11 inches, the thickness of the double convex crown lens inch at the edge, and that of the concave flint lens 1 inch, we must take the radii of the surfaces in proportion to the focal length, and thus have

r1 = 12.74311 feet,

r2= 5.80716,

7327.15803,

Su=.03 nearly, from Sir D. Brewster's tables.

With these we obtain, by substituting in the formula before investigated, as follows:

S(D-D')=y1.du' {t,. 0009555 + t2.00006055}

=.000000501759375.

To find the diameter of the least circle of dispersion,

Let RM, be as in the last figure, (see Fig. 5.) qRq the angle of residual dispersion

=8 (D-D');

then ab, the diameter of the least circle into which all the coloured rays are collected,

=qq' ×

R3 M3 9 M 3

;

and in the triangle qR.q', we have qq' = 9 R ̧ . sin q Rəq′

sin Req' M3

3

and thus, in our example, the diameter of circle of residual dispersion

=.000009533 of a foot.

The angle which this subtends at the object-glass is (D-D'); and, measured in seconds of a degree, becomes

.0000005017
.0000048481

= 0".103 nearly.

A residual circle of dispersion of this magnitude is such as would never be tolerated in a telescope like the Northumberland one, from which we have taken our example. The observations made by Professor Struve with the Dorpat telescope, on most difficult double stars, shew that no uncorrected dispersion to an amount like the above could exist in that telescope; and the Northumberland telescope may be reasonably expected to be no ways inferior.

From this we are led to conclude, that practical opticians have through experience adopted curvatures for their lenses of much greater accuracy than those given by any theoretical computations hitherto published, and the production of critical defining power in an objectglass must be left to their skill and patience in finding the forms which produce the desired effect.

To shew the effect of our correction on the radius of any one of the surfaces, I shall now give, as example, a case in which the convex crown lens is taken of a greater thickness than would occur in any modern object-glass, namely

t1 = inch, tinch for an aperture of 6 inches, and focal length 10 feet.

Calculating with these, we find

S(D-D')=.0000011629,

and the diameter of the least circle of residual dispersion

=.000011629 of a foot.

Now the diameter of the least circle of spherical aberration in a crossed lens of plate-glass, refractive index = 1.5, is for the same focal length and aperture

= .00008370;

so that the former correction would amount to about one-seventh of the spherical aberration in an equivalent lens of the best form, and yet to correct this large residual dispersion would require only a very small alteration in the radius of one of the surfaces. To find this alteration, we must now take the value of (D-D')=0, and as both the terms in the residual value are positive, we cannot fulfil this conδ бл dition whilst is separately = 0, but must make the whole

expression =0, when the values of t and t2 are given for any particular lens. I have performed the calculation to this effect for the above example, supposing the last surface of the flint-lens only to be altered, and I find that its radius (r) must be increased by 30 1th of an inch nearly, which is a very small quantity in 14 feet, and shews us that it is scarcely to be hoped, that we can obtain a very fine objectglass by trusting to theoretical computations solely; but that, after the general forms have been investigated for the optician, we must rely on his experience to vary his curvatures slowly until he has obtained the maximum effect of distinctness.

With respect to the actual thicknesses adopted in England, I am indebted for information to the liberality of Mr Tully and Mr Robinson; and as it is important that such information should be recorded in print, I shall not hesitate to give the full extract from Mr Robinson's reply to my letter to him requesting such information. He says: "Thinking that the best information might be obtained from Mr Tully, I called on him, and not being so fortunate as to find him at home, I left your letter, with the request that he would furnish the information you required; he has just now called on me and tells me, that there is no absolute rule for the thickness of either the concave or convex lens; that great thickness for the convex lens, if it be of crown glass, is considered objectionable on account of its colour occasioning loss of light; and its being thin is objected to, but merely because if it be ground to a sharp edge, there is danger of the edge being broken in polishing: he has just made an object-glass of 51 inches diameter, the thickness of the crown glass is at the edge of an inch, and that of the concave, and these he thinks very proper thicknesses and would not wish they should be thicker; but had these disks been thinner, they, (being of good glass,) would not have been rejected on that account; and in general, the only rule for thickness is, that it be such that the edge of the convex be not splintered in working, and the centre of the concave be not so thin as to change its form in polishing."