and is greater for flint than for crown glass. If two prisms of these substances, of small refracting angles, be combined into one, with their edges turned opposite ways, they will achromatize one another if (μ” – μ) A, or the product of deviation by dispersive power, is the same for both. As the deviations can be made to have any ratio we please by altering the angles of the prisms, the condition is evidently possible.

The deviation which a simple ray undergoes in traversing a lens, at a distance x from the axis, is, f denoting the focal length of the lens (§ 1004), and the separation of the red and violet constituents of a compound ray is the product of this deviation by the dispersive power of the material. If a convex and concave lens are combined, fitting closely together, the deviations which they produce in a ray. traversing both, are in opposite directions, and so also are the dispersions. If we may regard x as having the same value for both (a supposition which amounts to neglecting the thicknesses of the lenses. in comparison with their focal lengths) the condition of no resultant dispersion is that

dispersive power x

1 f

has the same value for both lenses. Their focal lengths must therefore be directly as the dispersive powers of their materials. These latter are about 033 for crown and 052 for flint glass. A converging achromatic lens usually consists of a double convex lens of crown fitted to a diverging meniscus of flint. In every achromatic combination of two pieces, the direction of resultant deviation is that due to the piece of smaller dispersive power.

The definition above given of dispersive power is rather loose. To make it accurate, we must specify, by reference to the "fixed lines," the precise positions of the two rays whose separation we consider.

Since the distances between the fixed lines have different proportions for crown and flint glass, achromatism of the whole spectrum is impossible. With two pieces it is possible to unite any two selected rays, with three pieces any three selected rays, and so on. It is considered a sign of good achromatism when no colours can be brought into view by bad focussing except purple and green.

1065. Achromatic Eye-pieces.-The eye-pieces of microscopes and astronomical telescopes, usually consist of two lenses of the same kind of glass, so arranged as to counteract, to some extent, the spherical

and chromatic aberrations of the object-glass. The positive eye-piece, invented by Ramsden, is suited for observation with cross-wires or micrometers; the negative eye-piece, invented by Huygens, is not adapted for purposes of measurement, but is preferred when distinct vision is the sole requisite. These eye-pieces are commonly called achromatic, but their achromatism is in a manner spurious. It consists not in bringing the red and violet images into true coincidence, but merely in causing one to cover the other as seen from the position occupied by the observer's eye.

In the best opera-glasses (§ 1033), the eye-piece, as well as the object-glass, is composed of lenses of flint and crown so combined as to be achromatic in the more proper sense of the word.

1066. Rainbow.-The unequal refrangibility of the different elementary rays furnishes a complete explanation of the ordinary phenomena of rainbows. The explanation was first given by Newton, who confirmed it by actual measurement.

It is well known that rainbows are seen when the sun is shining on drops of water. Sometimes one bow is seen, sometimes two, each of them presenting colours resembling those of the solar spectrum. When there is only one bow, the red arch is above and the violet below. When there is a second bow, it is at some distance outside of this, has the colours in reverse order, and is usually less bright.

Rainbows are often observed in the spray of cascades and fountains, when the sun is shining.

In every case, a line joining the observer to the sun is the axis of the bow or bows; that is to say, all parts of the length of the bow are at the same angular distance from the sun.

The formation of the primary bow is illustrated by Fig. 765. A ray of solar light, falling on a spherical drop of water, in the direction SI, is refracted at I, then reflected internally

from the back of the drop, and again refracted into the air in the

direction I'M. If we take different points of incidence, we shall obtain different directions of emergence, so that the whole light which emerges from the drop after undergoing, as in the figure, two refractions and one reflection, forms a widely-divergent pencil. Some portions of this pencil, however, contain very little light. This is especially the case with those rays which, having been incident nearly normally, are returned almost directly back, and also with those which were almost tangential at incidence. The greatest condensation, as regards any particular species of elementary ray, occurs at that part of the emergent pencil which has undergone minimum deviation. It is by means of rays which have undergone this minimum deviation, that the observer sees the corresponding colour in the bow; and the deviation which they have undergone is evidently equal to the angular distance of this part of the bow from the sun.

The minimum deviation will be greatest for those rays which are most refrangible. If the figure, for example, be supposed to represent the circumstances of minimum deviation for violet, we shall obtain smaller deviation in the case of red, even by giving the angle I AI′ the same value which it has in the case of minimum deviation for violet, and still more when we give it the value which corresponds to the minimum deviation of red. The most refrangible colours are accordingly seen furthest from the sun. The effect of the rays which undergo other than minimum deviation, is to produce a border of white light on the side remote from the sun; that is to say, on the inner edge of the bow.1

1 When the drops are very uniform in size, a series of faint supernumerary bows, alternately purple and green, is sometimes seen beneath the primary bow. These bows are produced by the mutual interference of rays which have undergone other than minimum deviation, and the interference arises in the following way. Any two parallel directions of emergence, for rays of a given refrangibility, correspond in general to two different points of incidence on any given drop, one of the two incident rays being more nearly normal, and the other more nearly tangential to the drop than the ray of minimum deviation. These two rays have pursued dissimilar paths in the drop, and are in different phases when they reach the observer's eye. The difference of phase may amount to one, two, three, or more exact wave-lengths, and thus one, two, three, or more supernumerary bows may be formed. The distances between the supernumerary bows will be greater as the drops of water are smaller. This explanation is due to Dr. Thomas Young.

A more complete theory, in which diffraction is taken into account, is given by Airy in the Cambridge Transactions for 1838; and the volume for the following year contains an experimental verification by Miller. It appears from this theory that the maximum of intensity is less sharply marked than the ordinary theory would indicate, and does not correspond to the geometrical minimum of deviation, but to a deviation sensibly greater. Also that the region of sensible illumination extends beyond this geometrical minimum and shades off gradually.

The condensation which accompanies minimum deviation, is merely a particular case of the general mathematical law that magnitudes remain nearly constant in the neighbourhood of a maximum or minimum value. The rays which compose a small parallel pencil SI incident at and around the precise point which corresponds to minimum deviation, will thus have deviations which may be regarded as equal, and will accordingly remain sensibly parallel at emergence. A parallel pencil inci

dent on any other part of the drop, will be divergent at emergence. The indices of refraction for red and violet rays from air into water are respectively

108 and 102, and calculation shows that the distances from the centre of the sun to the parts of the bow in which these colours are strongest should be the. supplements of 42° 2' and 40° 17' respectively. These results agree with observation. The angles 42° 2′ and 40° 17' are the distances from the antisolar point, which is always the centre of the bow.

The rays which form the secondary bow have

S

undergone two internal reflections, as represented in Fig. 766, and here again a special concentration occurs in the direction of minimum deviation. This deviation is greater than 180° and is greatest for the most refrangible rays. The distance of the arc thus formed from the sun's centre, is 360° minus the deviation, and is accord

ingly least for the most refrangible rays. Thus the violet arc is nearest the sun, and the red furthest from it, in the secondary bow.

Some idea of the relative situations of the eye, the sun, and the drops of water in which the two bows are formed, may be obtained from an inspection of Fig. 767.

SUNDRY ADDITIONS TO PREVIOUS CHAPTERS.

1066A. Goniometers.-A goniometer is an instrument for measuring the angle between two plane faces either of a crystal or of a prism. The measurement is usually made by means of reflections from the two faces. This may be done in either of the two following ways. For convenience of description we shall assume that the edge in which the two faces meet is vertical; in practice it may have any direction.

First method.-Observe in one of the two faces the reflection of an object at a few yards' distance, in the same horizontal plane with the prism; and by rotating the prism in this plane bring the image into apparent coincidence with some other object; or, if preferred, bring it upon the cross-wires of a fixed telescope. Then rotate the prism in the horizonal plane till the other face gives an image of the same object in the same position. The second face is now in or parallel to the position previously occupied by the first face, and the angle through which the prism has been turned is the angle between one face and the other face produced. By subtracting it from 180° we obtain the required angle between the faces. The goniometer is furnished with a graduated circle on which the rotation is read off.

Second method. The goniometer must have a telescope (with

aa

cross-wires) which can travel round the graduated circle, while always directed towards its centre, where the prism stands. The prism is placed in such a position that rays from a distant object, or more conveniently from a slit in the focus of a collimating lens, fall upon both faces simultaneously. The telescope is first placed so as to receive on its cross-wires the image formed by reflection at one face, and is then moved past the base of the prism till it receives the image formed by reflection at the other face. The angle through

Fig. 767A.-Measure ment of Angle of

Prism.