In order to state the laws of the phenomena with precision, it is necessary to attend to the crystalline form of Iceland-spar.

At the corner which is represented as next us in Fig. 711 three equal obtuse angles meet; and this is also the case at the opposite

corner which is out of sight. If a line be drawn through one of these corners, making equal angles with the three edges which meet there, it or any line parallel to it is called the axis of the crystal; the axis being properly speaking not a definite line but a definite direction.

The angles of the crystal are the same in all specimens; but the

lengths of the three edges (which may be called the obique length, breadth, and thickness) may have any ratios whatever. If the crystal is of such proportions that these three edges are equal, as in the first part of Fig. 713, the axis is the direction of one of its diagonals, which is represented in the figure.

Any plane containing (or parallel to) the axis is called a principal plane of the crystal.

If the crystal is laid over a dot on a sheet of paper, and is made

to rotate while remaining always in contact with the paper, it will be observed that, of the two images of the dot, one remains unmoved, and the other revolves round it. The former is called the ordinary, and the latter the extraordinary image. It will also be observed that the former appears nearer than the latter, being more lifted up by refraction.

The rays which form the ordinary image follow the ordinary law of sines (§ 983). They are called the ordinary rays. Those which form the extraordinary image (called the extraordinary rays) do not follow the law of sines, except when the refracting surface is parallel to the axis, and the plane of incidence perpendicular to the axis; and in this case their index of refraction (called the extraordinary index) is different from that of the ordinary rays. The ordinary index is. 1.65, and the extraordinary 1:48.

When the plane of incidence is parallel to the axis, the extraordinary ray always lies in this plane, whatever be the direction of the refracting surface; but the ratio of the sines of the angles of incidence and refraction is variable.

When the plane of incidence is oblique to the axis, the extraordinary ray generally lies in a different plane.

We shall recur to the subject of double refraction in the concluding chapter of this volume.

CHAPTER LXX.

LENSES.

1000. Forms of Lenses.-A lens is usually a piece of glass bounded by two surfaces which are portions of spheres. There are two principal classes of lenses.

1. Converging lenses or convex lenses, which have one or other of the three forms represented in Fig. 714. The first of these is called double convex, the second plano-convex, and the third concavoconvex. This last is also called a converging meniscus. All three

Fig. 714.-Converging Lenses.

Fig. 715.-Diverging Lenses.

are thicker in the middle than at the edges. They are called converging, because rays are always more convergent or less divergent after passing through them than before.

2. Diverging lenses or concave lenses (Fig. 715) produce the opposite effect, and are characterized by being thinner in the middle than at the edges. Of the three forms represented, the first is double concave, the second plano-concave, and the third convexo-concave (also called a diverging meniscus).

From the immense importance of lenses, especially convex lenses, in practical optics, it will be necessary to explain their properties at some length.

1001. Principal Focus.-A lens is usually a solid of revolution, and the axis of revolution is called the axis of the lens, or sometimes the principal axis. When the surfaces are spherical, it is the line joining their centres of curvature.

When rays which were originally parallel to the principal axis

called the principal focus. The distance A F of the principal focus from the lens is called the principal focal distance, or more briefly and usually, the focal length of the lens. cipal focus at the same distance on the

There is another prinother side of the lens, corresponding to an incident beam coming in the opposite direction. The focal length depends on the convexity of the surfaces of the lens, and also on the refractive power of the material of which it is composed, being shortened either by an

increase of refractive power or by a diminution of the radii of curvature of the faces.

In the case of a concave lens, rays incident parallel to the principal axis diverge after passing through; and their directions, if produced backwards, would approximately meet in a point F (Fig. 717), which is still called the principal focus. It is only a virtual focus, inasmuch as the emergent rays do not actually pass through it, whereas the principal focus of a converging lens is real.

1002. Optical Centre of a Lens. Secondary Axes.-Let O and O' (Fig. 718) be the centres of the two spherical surfaces of a lens. Draw any two parallel radii O I, O'E to meet these surfaces, and let the joining line I E represent a ray passing through the lens. This ray makes equal angles with the normals at I and E, since these latter are parallel by construction; hence the incident and emergent rays SI, ER also make equal angles with the normals, and are therefore parallel. In fact, if tangent planes (indicated by the dotted lines in the figure) are drawn at I and E, the whole course of the ray SIER will be the same as if it had passed through a plate bounded by these planes.

Let C be the point in which the line IE cuts the principal axis, and let R, R' denote the radii of the two spherical surfaces. Then, from the similarity of the triangles OCI, O'CE, we have

line of centres O O' in a Every ray whose direc

which shows that the point C divides the definite ratio depending only on the radii. tion on emergence is parallel to its direction before entering the lens, must pass through the point C in traversing the lens; and conversely, every ray which, in its course through the lens, traverses the point C, has parallel directions at incidence and emergence. The point C which possesses this remarkable property is called the centre, or optical centre, of the lens.

In the case of a double convex or double concave lens, the optical centre lies in the interior, its distances from the two surfaces being directly as their radii. In plano-convex and plano-concave lenses it is situated on the convex or concave surface. In a meniscus of either kind it lies outside the lens altogether, its distances from the surfaces being still in the direct ratio of their radii of curvature.1

1 These consequences follow at once from equation (1); for the distances of C from the