paths from S to corresponding points in consecutive apertures is, as before, (a + b) sin i. In the same way, the optical

differences of the paths from these points to P is (a + b) sin 0. Thus we get maximum illumination at P if (d+b)(sin i + sin 0) is a multiple of λ.

These grating images may be seen without any very special apparatus. If a small source of light be looked at through a fine silk handkerchief, a few images will be seen on each side of the source. They may be seen on looking at a gaslight through the silk cover of an umbrella. Very fine wire gauze will also produce them. In these cases the retina of the eye is the screen on which the images are thrown, and the lens of the eye takes the place of the lens L'.

Suppose white light to be sent through the slit to the grating, with the arrangement of lenses just explained. Then each separate colour will give rise to a series of diffraction images. In any one set of images of the various colours the separate coloured images will be arranged side by side, the positions of the images varying continuously with the wavelengths, those of longer wave-lengths being less deviated than those of shorter. Thus we shall get a series of coloured spectra. In any one spectrum the position of any colour will depend on the wave-length of the colour, the number of lines per unit length of the grating, and the arrangement of the apparatus. It will not depend upon the nature of any material used, as the relative positions of the colours in a

prismatic spectrum depend upon the nature of the glass. If the incident light falls normally on the grating, we have the simple law, that in any spectrum the sine of the angle of deviation of any colour from the direct image of the slit is simply proportional to the wave-length of that colour. For this reason, grating spectra are frequently used as standards of reference rather than prismatic spectra.

The optical grating affords the most accurate method of measuring the wave-length of light of a given quality. For this purpose it is used with a spectrometer. The slit of the collimator is illuminated with light of the given quality. The collimator and telescope of the spectrometer are set for parallel light, and the grating is set on the table of the spectrometer with its lines normal to the table. To do this, first get the face normal. This may be tested, as in setting the prism on the spectrometer, by observing whether the reflected image of the slit is formed at the proper level. Next, if necessary, turn the grating in its own plane, to get the lines right. This adjustment is complete when the diffraction images are all formed at the same level. The measurements may now be made in either of two ways.

1. Set the grating normal to the incident light. Set the telescope to view any image, say the nth, to right and to left of the direct image. Half the difference between the readings of the telescope is the angular deviation of the image. Call this deviation On

The number of lines of the grating per unit of length (say per millimetre) is measured. This may be done by means of a microscope, using the micrometer eye-piece and a standard micrometer scale. Let this number be N. Then we have, if A is the required wave-length

sin On

= nNλ.

2. If the normal to the grating makes an angle i with the incident light on one side, and an angle with the light going to the nth image on the other, we have

The angular deviation of the nth image from the light coming from the slit is in this case i +0. Then this deviation

i - 0

is least when cos is greatest, that is, when i = 0. If,

2

then the grating is set to produce the nth image in the

position of minimum deviation, and this deviation is „, since Pn= i +0, we have

The angle may be determined, as before, by observing the nth image to right and to left of the direct image.

Reflexion Grating. Suppose a plane reflecting metallic surface to be ruled with a large number of fine equidistant parallel lines. This will produce diffraction images similar to those of the transparent grating. Waves falling on the reflecting grating will travel out in all directions on the side of the incident light, and in certain directions will produce very

FIG. 163.

strong effects. These directions may be found in just the same manner as that already employed. Suppose there are N lines per unit length of the grating. Let the incident light make an angle, i, with the normal. Suppose that a maximum of effect is produced by the irregularly reflected light in a direction making an angle, 0, with the normal on the same side. Let A and B be corresponding points on two of the reflecting lines. Draw B M, B N perpendicular to the incident and irregularly reflected light at A. Then a maximum of effect is produced in the direction 0, by the united action of corresponding points of the various reflecting strips, when MAAN is a multiple of the wave-length, that is, since

If the angle is measured on the other side of the normal, the condition is

The greatest effect is produced when all the points of all the reflecting strips act together, that is, in the direction of regular reflexion.

Curved gratings have been used by Professor Rowland, and by Professor Langley in his researches on solar radiation. Imagine a concave reflecting surface ruled with lines; and let the plane of the paper represent a plane cutting the reflecting strips normally at A, B, C. Suppose S is a source of light, and P a point such that the paths SAP, SBP, SCP, . . . differ by a constant multiple of a wave-length. Then an image of S will be formed at P. This image can be thrown on a screen or photographed without the assistance of a lens.

If

B

A

FIG. 164.

S

light of mixed quality be used, there will be a series of images such as P formed by the lights of various colours. Thus a spectrum is formed by means of a narrow slit placed at S. By directing various parts of the solar spectrum so formed into suitable apparatus, and especially of that part of it which is below the red, and is formed of rays whose waves are too long to produce the effect of light, Professor Langley has made his celebrated researches on this, the "infra-red" solar spectrum.

Some substances of striated structure, such as mother-ofpearl, show beautiful colours on account of acting on the incident light as reflexion gratings.

Circular Aperture. Suppose light comes from a small source, S, and a screen with a circular aperture of small radius is placed so that S is on the axis of the circle. Let the radius

of the circle be r. Let the distance of S from the circle be a. To consider the illumination at a point on the axis of the

FIG. 165.

circle, at a distance b from the centre, and on the side opposite to S.

Take C, the centre of the circle. Let Q be any point of it at distance x from C.

Suppose the area of the circle divided up into a large number, n, of zones equal in area, and each having centre at C. The area of each is

n

Each zone being very narrow, we

may suppose that the vibrations from O passing through all points of it, reach P in the same phase. The displacement at P produced by disturbances reaching it through the zone of mean radius x, may be written

If this is the mth zone from the centre, x2 = m.

thus we write the above displacement—