known law which will be proved in Art. 75, the magnifying power of a telescope is equal to the ratio of the widths of the incident and emergent beams. If the width of the emergent beam is greater than the greatest width p which is capable of entering the pupil of the eye, the full aperture is not made use of. Hence to obtain full resolving power the magnifying power of a telescope should be not less than Dp. If it is less, the rays entering the outer portion of the telescope lens do not enter the eye at all and may as well be blocked out altogether, thus reducing the aperture to its useful portion.

72. Resolving Power of the Eye. We may apply equation (4) to the case of two stars or other point sources being looked at directly by the eye. An apparent complication arises owing to the fact that the wave-length of light in the vitreous humour, which is the last medium through which it passes, is not the same as the wave-length in air, but this makes no difference provided that we take the width of the beam as above defined. Let Fig. 95 represent

for

A

B

Ρ

W

1

Fig. 95.

diagrammatically a beam of light entering the media of the eye. If a plane wave-front passes through an aperture AB of such size that the beam passing through it may just enter the pupil of the eye, the first dark ring of the diffraction images passes through Q when the difference in optical lengths from A to Q exceeds by 1.22λ that from B to Q. Also a wave-front parallel to W'F" has the centre of its diffraction image at Q when the optical distance from all points of its plane to Q is the same, hence AT must be equal to 1.22λ, and the angle between AB and W'F" is measured by AT/AB or

Here is the wave-length measured in air.

The width of pupil is variable, but with light of medium intensity such that p is about 3 mm. (the actual opening of the pupil will be less, owing to the convergence produced by the cornea), two small point sources of light should be resolvable by the eye when at an angular distance of 42". Helmholtz gives for the experimental value of the smallest angular distance perceptible by the eye the range between 1' and 2', which would show that with full aperture of the pupil, our sense of vision is limited rather by the optical defects of the eye and physiological causes than by diffraction effects.

73. Rectangular Apertures. If the surface of the telescope is covered by a diaphragm having a rectangular aperture, the distribution of light is more easily calculated, and may be expressed accurately in a

simple form. Take the axis of x and y in the focal plane of the lens and parallel to the sides of the aperture and let the length and width of the aperture in the direction of x and y be a and b respectively.

When b is large we obtain the case investigated in Art. 53, where the intensity was found to be proportional to (sin2 a)/a2 where a=πax/ƒλ. If a is large, the expression must be proportional to (sin2 ß)/ẞ2 where B = by fλ. We can satisfy both conditions if in the general case we take the intensity to be proportional to sin2 a sin2 ß/a2ß2. The constant to be applied may be found by considering that if a and b are very small, the amplitude must by Art. 46 be equal to ab/fλ if the incident beam has unit intensity. Hence for the complete expression we obtain

The most important case we shall have to consider is that where the source is a luminous line parallel to the direction of y. A point of the luminous line at a distance y' from its central point causes an intensity at the central point of the image which may be obtained from (5) by making the value of y contained in ẞ equal to y'e/e' where 2e' is the length of the source and 2e that of its image. Hence the total intensity at the point y = 0 is proportional to

Ife is large, we may substitute infinity for the limits, and as

it follows that the intensity in the image is proportional to

where a has the same value as before. It follows that the total amount of energy which is transmitted in unit time through a small surface s of the image is κSI', where κ is a constant which may be determined as follows. If a ribbon of unit width be cut out transversely to the image, the total amount of energy transmitted through the ribbon is

K

If E denote the amount of light from unit length of the source transmitted through unit surface of the first lens, and m the magnifying power, the total amount of light per unit length of the image is mEab. Hence K= mE.

74. Luminous Surfaces. The image of a surface bounded by a straight edge may be calculated from the above. Dividing the

surface into narrow strips parallel to one of the edges, each strip will have a diffraction image according to Fig. 70, and at each point of the

image we should have to add up the effect due to each strip. It is easy to see that at the geometrical image of the edge, the intensity is half that observed at some distance inside the edge, where the illumination is uniform, for when two similar surfaces are placed against each other with their edges AB in contact, a uniformly illuminated sheet is obtained, and each half must contribute equally to the illumination at the dividing line. The intensity at other points can only be expressed in the form of definite integrals or calculated by means of a series. The intensities are plotted in Fig. 96. The dotted line AB marks the edge of the geometrical image of the surface. The intensity at that point is 5, and falls off rapidly towards the outside of the image.

When a telescope is used to examine such a surface as the moon, it is not a question of separating two luminous points or sharply defined surfaces, but rather of interpreting changes of luminosity in a continuously varying surface. Details which are as near together as two stars when at the point of optical separation may be indistinguishable on an illuminated surface. If we double that distance the central diffraction bands stand altogether clear of each other, and hence the angular distance between two points should be equal to 2:44A/D, if there is to be no overlapping at all. The edge of the image of a luminous surface is not bounded by alternately bright and dark fringes, and there is no definite boundary at which the image of the surface can be said to end.

For a given distance from the geometrical edge the intensity is less than at the same distance from the image of a narrow aperture. Hence, as has been pointed out by Wadsworth, the images of two surfaces

may be put closer together than the images of the slit without their images becoming confused.

The points marked and 2 on the horizontal line of Fig. 96 represent the places where the first two minima of light would occur in the image of a narrow slit coincident with AB.

75.

Illumination of the image of a luminous surface. The

resultant energy which leaves a luminous surface is the same in all directions for equal cross-sections of the beam. As with a given small surface S Fig. 97 Fig. 97. the cross-section of the beam varies as the cosine of the direction angle, the intensity of radiation sent out by a surface S is proportional to cos 0, but for small inclinations to the normal, we may take the radiation to be independent of the direction. If an image of a surface S is to be formed, the illumination of the image must be proportional to the amount of light which the luminous surface sends through the optical system. If all the light which passes through the first lens passes also through the other lenses, this is proportional to the surface S, and to the solid angle w subtended by the lens at a point of S. We may therefore write for the light passing through the optical system ISo, where I solely depends on the luminosity of the surface. If s is the size of the image of S, and if the image is such that the illumination is uniform, the brightness of the image is equal to ISw/s.

W

We shall first consider the case that the linear dimensions of s are such that the diameter of the diffraction disc may be neglected in comparison with it, so that we may find the relation between S and s by the laws of geometrical optics. Let LL and MM' be the

RL

wave-fronts diverging from P and

converging to the image respectively, and imagine a second

Fig. 98.

wave-front RR' slightly inclined to the first, to diverge from P'. If POP'O, the second wave-front may be obtained by turning LL' about / through a small angle 0. The optical length from P' to L has been increased by the change, by the quantity RL, and the optical length from P' to L' has diminished by the same amount. The optical lengths from L to M and L'M' have not been altered (Art. 23). Hence if Q' is the image of P' the optical length M'Q' must differ from MQ' by 2RL, the total length from P' to Q' being the same whether measured by way of LM or by way of L'M'. It follows that to obtain Q' we must turn round the wave-front M'M through such an angle that HM=2RL. If D is the width of the beam at LL' and d the

width at MM', the angles POP' and QOQ' are 2RL/D and MH|d respectively, and are therefore in the inverse ratio of D: d. It follows that

If a square of surface S and sides P, P' is formed in a plane at right angles to 00', its image S will be a square with Q, Q' as sides, hence

The solid angle (w) of the beam entering the first lens is D2/4P02, and the solid angle (w) of the beam converging to Q is d/4O'Q. Hence the illumination per unit surface of s is

Before discussing the last equation, we note two interesting results which have incidentally been obtained in the investigation.

QQ' is inverted as compared with PP' and this must always be the case according to the construction when the limiting ray MQ is the continuation of the ray PL on the same side of the axis, but if the rays have crossed once or an odd number of times between 0 and O', so that the ray PL becomes the ray M'Q, we should have to turn round the ultimate wave-front MM' in the opposite direction in order to equalize the optic lengths of the extreme rays, and the image would then be erect.

The ratio of the angles QO'Q' and POP' becomes the magnifying power (m) of the arrangement, when, as in a telescopic system, the incident and emergent beams are both parallel, hence

which proves the proposition which has already been made use of in Art. 71.

The theorem, defined by equation (7), that the brightness of a luminous surface is determined by the solid angle of the converging pencil which forms the image, is of fundamental importance. We may derive three separate conclusions from it. (1) The apparent brightness of a luminous surface looked at with the naked eye is independent of its distance from the observer. (2) No optical device can increase the