due to dispersion, as far as possible; so that we must choose which of the two corrections should be effected, and which left.

It is then usual to make the coloured images have the same magnitude; for the eye is a better judge of the magnitude of an object than of its distance.

Using the same notation as before, the condition is that B/B should be the same for the two colours corresponding to μ, μ.

For the ray of refractive index u', this becomes

x (1+w) (x + a) (1+w') ax (1+) (1+ w')

+

ff'

This is therefore the condition for the partial achromatism of the two lenses. In general, it is not independent of the position of the object.

99. If we consider the inclinations of rays to the axis of the instrument, instead of the magnifying power, it will be seen that we have ensured that two differently coloured rays diverging from the object will emerge parallel to each other.

For if a, a' be the inclinations to the axis of the original and final rays, cutting the axis at the points determined by x, y, we may see directly from a figure, or by Helmholtz' theorem relating to the magnifying power, that

tan a' xy'
tan a x'y

so that if the condition previously found be satisfied, then a' is the same for the two colours; and the final rays emerge parallel to each other.

100. The most useful application of this condition is to the achromatism of eye-pieces. The rays strike the eye-piece excentrically diverging from the image formed by the object-glass. The images formed by the lenses of the eye-pieces are formed exactly as if the rays diverged from a real object, except that the rays from any point of the image do not fill the whole of the lens.

The centre of the object-glass is usually very distant as compared to the focal lengths of the lenses of the eyepiece. If we make x very large in the previous equation of condition, it becomes

This condition may be derived in a shorter manner for this particular case by making the focal length of the equivalent lens the same for two colours.

There is a special advantage in making the lenses of the same kind of glass, because then if we make two coloured images coincide, all the coloured images will be united. The condition for achromatism then becomes

or in words, the distance between the lenses must be half the sum of their focal lengths.

EXAMPLES.

1. Shew that at a single refraction at a plane surface the dispersion is proportional to the tangent of the angle of refraction.

2.

The refractive index of a medium for the two rays at the red and violet ends of the spectrum being 1.63 and 1.66 respectively, calculate the dispersive power. Ans. .

3. Calculate the dispersive power of a medium for which the refractive indices for the same two rays are 1.53 and 1.54 respectively, and find the ratio between the focal lengths of two lenses formed of the media in this and the last example, that the combination may be achromatic when the lenses are placed in contact. Ans. 187, 43: 107.

4. Prove that if ƒ be the focal length of a lens, w its dispersive power, the distance from the centre of the lens of the point to which a pencil of standard rays is made to converge, the distance between the foci of the red and violet rays for the same incident ray is approximately wvf.

5. The dispersive power of a medium is 036. The focal length of a lens formed of it being 3 feet for standard rays, find the distance between the extreme images of the sun formed by the lens. Ans. 0108 feet.

6. If μ, v be the indices of refraction for the red and violet rays, respectively, for crown-glass, and ', ' be the indices for the same rays for flint-glass; and if two thin lenses be constructed, one double convex of crown-glass with each surface of radius r, and one double concave of flint-glass with its surfaces of radii r and s, and they be placed in contact so that the light is incident on the surface of radius s; then the combination will be achromatic if

7. A small pencil of parallel rays of white light, after transmission in a principal plane through a prism, is received on a screen whose plane is perpendicular to the direction of the pencil; prove that the length of the spectrum will be proportional to

(μv — μr) sin÷cos2 D cos (D+1−4) cos p';

where is the refracting angle, o, o' the angles of incidence and refraction at the first surface, and D the deviation of the mean ray.

8. If an achromatic eye-piece for an astronomical telescope be composed of two convex lenses of different materials, prove that the distance between them must be intermediate between ƒ' and lf/(l-ƒ), where ƒ is the absolute focal length of the field-glass, f' that of the eye-glass, and 7 the length of the telescope from object-glass to field-glass.

9. Prove that a system of three thin convex lenses made of the same material, placed so that the distance between the first and second is a, and that between the second and third is b, is achromatic for a pencil coming from a point on the axis whose distance from the first lens is

2abf-af1f3-(a+b) fif2
3ab-2bf1-2(a+b)f2-2af3+f2f3+f3f1+ff2

where f1, f2, fa are the focal lengths (taken positively) of the three lenses, respectively.

CHAPTER VII.

THE EYE, AND VISION THROUGH LENSES.

101. THE eye is an optical instrument consisting essentially of a series of refracting media bounded by curved surfaces, and a delicate network of small nerve-fibres forming part of the optic nerve; a pencil of light incident upon the eye is refracted at the curved surfaces and brought to a focus on the network of nerve-fibres, and the impression is carried to the brain along the optic nerve.