Optics

OPTICS, Geometrical. The subject of optics is so extensive that some subdivision of it is convenient if not necessary. Under the head of Light will be found a general sketch accompanied by certain developments. The wave theory and those branches of the subject which are best expounded in connexion with it are reserved for treatment in a later volume. The object of the present paper is to give some account of what is generally called geometrical optics,—a theoretical structure based upon the laws of reflexion and refraction. We shall, however, find it advisable not to exclude altogether the conceptions of the wave theory, for on certain most important and practical questions no conclusions can be drawn without the use of facts which are scarcely otherwise interpretable. Indeed it is not to be denied that the too rigid separation of optics into geometrical and physical has done a good deal of harm, much that is essential to a proper comprehension of the subject having fallen between the two stools.

Systems of Rays in General.— In the investigation of this subject a few preliminary propositions will be useful.

If a ray AB (fig. 1) travelling in a homogeneous medium suffer reflexion at a plane or curved surface BD, the total path between any two points A, C on the ray is a minimum, t.e., AB + BC is less along the actual path than it would be if the point B were slightly varied.

For a variation of B in a direction perpendicular to the plane of reflexion (that of the diagram) the truth of this statement is at once evident. For a small variation BB' in the plane of reflexion we see that the difference AB' - AB is equal to the projection of BB' upon AB, and that the difference CB - CB' is equal to the projection of BB' upon BC. These projections are equal, since by the law of reflexion AB and BC are equally inclined to BB', and thus the variation of the total path, AB'+ B'C - (AB + BC), vanishes.

A corresponding proposition holds good in the case of refraction. If we multiply the distances travelled in the first and second media respectively by the refractive indices appropriate to the media, the quantity so obtained is a minimum for the actual path of the ray from any point to any other. It is sufficient to consider the case of a variation of the point of passage in the plane of refraction.

In the first medium (fig. 2) μ AB' - μ. AB = μBB'cos ABD, and in the second medium μ'OB - μ'OB' = μ'BB' cos CBD. The whole variation of the quantity in question is therefore

BB' (μ cos ABD - μ cos CBD). Now by the law of refraction the sines of the angles of incidence and refraction are in the ratio μ' : μ, and accordingly μ cos ABD - μ cos C’BD = 0.

In whichever direction, therefore, the point of transition be varied, the variation of the quantity under consideration is zero. It is evident that the second proposition includes the first, since in the case of reflexion the two media are the same.

The principle of the superposition of variations now allows us to make an important extension. If the quantity, which we may denote by ∑μs, be a minimum for separate variations of all the points of passage between contiguous media, it is also a minimum even when simultaneous variations are admitted. However many times a ray may be reflected or refracted at the surfaces of various media, the actual path of the ray between any two points of its course makes ∑μs a minimum. Even if the variations of refractive index be gradual instead of sudden, the same principle holds good, and the actual path of the ray makes Jμds, as it would now be written, a minimum.

The principle itself, though here deduced from the laws of reflexion and refraction, is an immediate consequence of the fundamental suppositions of the wave-theory of light, and if we are prepared to adopt this point of view we may conversely deduce the laws of reflexion and refraction from the principle. The refractive index μ is inversely proportional to the velocity of propagation, and the principle simply asserts that in passing from any point to any other the light follows the shortest course, that is, the course of earliest arrival.

If two points be such that rays issuing from one of them, and ranging through a finite angle, converge to the other after any number of reflexions and refractions, the value of ∑μs from one focus to the other must be the same for all the rays.

Thus, in order to condense rays issuing from one point S upon a second point H by a single reflexion (fig. 3), the reflecting surface must be such that SP + HP = const., i.e., must be an ellipsoid of revolution with 8 and H for foci.

Again, if it be required to effect the same operation by a single refraction at the surface of a medium whose index is μ, we see that the surface (fig. 4) must be such that

SP + μ HP=const.

If S be at an in finite distance, i.e., if the incident rays be parallel, the surface is an ellipsoid of revolution with H for focus, and of eccentricity μ~ 1 (μ>l).

Another important proposition, obvious from the point of view of the wave-theory, but here requiring an independent proof, was enunciated by Malus. It asserts that a system of rays, emanating originally from a point, retains always the property of being normal to a surface, whatever reflexions or refractions it may undergo in traversing singly-refracting media.

Suppose that ABODE, A'B'C'D'E'. . . (fig. 5) are rays originally normal to a surface AA', which undergo reflexions or refractions at BB', CC', &c. On every ray take points E, E', &c., such that ~∑μs is the same along the courses AE, A'E', &c. We shall prove that the rays in the final medium are normal to the surface EE'.

For by hypothesis ∑μs along ABODE is the same as along A'B'C'D'E', and, by the property proved above to attach to every ray, ∑μs reckoned along the neighbouring hypothetical course A'BCDE' is the same as along A'B'C'D'E'. Hence ∑∕*s along A'BCDE' is the same as along ABODE, or (on subtraction of the common part) the same along A'B, DE' as along AB, DE. But since AB is perpendicular to AA', the value along A'B is the same as along AB, and therefore the value along DE' is the same as along DE; or, since the index is the same, DE = DE', that is, EE' is perpendicular to DE. The same may be proved for every point E' which lies infinitely near E, and thus the surface EE' is perpendicular to the ray DE, and by similar reasoning to every other ray of the system. It follows that reflexions and refractions cannot deprive a system of rays of the property of being normal to a surface, and it is evident that a system issuing from a point enjoys the property initially.

Consecutive rays do not in general intersect one another; but if we select rays which cut the orthogonal surface along a line of curvature, we meet with ultimate intersection, the locus of points thus determined being a caustic curve to which the rays are tangents. Other lines of curvature of the same set give rise to similar caustic curves, and the locus of these curves is a caustic surface to which every ray of the system is a tangent. By considering the other set of lines of curvature we obtain a second caustic surface. Thus every ray of the system touches two caustic surfaces.

In the important case in which the system of rays is symmetrical about an axis, the orthogonal surface is one of revolution. The first set of lines of curvature coincide with meridians. The rays corresponding to any one meridian meet in a caustic curve, and the surface which would be traced out by causing this to revolve about the axis is the first caustic surface. The second set of lines of curvature are the circles of latitude perpendicular to the meridians. The rays which are normal along one of these circles form a cone of revolution, and meet in a point situated on the axis of symmetry. The second caustic surface of the general theorem is therefore here represented by a portion of the axis.

The character of a limited symmetrical pencil of rays is illustrated in fig. 6, in which BAG is the orthogonal surface, and HFI the caustic curve having a cusp at F, the so-called geometrical focus. The distance FD between F and the point where the extreme ray BHDG cuts the axis is called the longitudinal aberration. On account of the symmetry FD is an even function of AB. If the pencil be small, we may in general consider FD to be proportional to AB2 , although in particular cases the aberration may vanish to this order of approximation. Let us examine the nature of the sections at various points as they may be exhibited by holding a piece of paper in the solar rays converging from a common burning-glass of large aperture. In moving the paper towards the focus nothing special is observed up to the position HI, where the caustic surface is first reached. A bright ring is there formed at the margin of the illuminated area, and this gradually contracts. At D the second caustic surface DF is reached, and a bright spot develops itself at the centre. A little farther back, at EG, the area of the illuminated patch is a minimum, and its boundary is called the least circle of aberration. Farther back still the outer boundary corresponding to the extreme rays begins to enlarge, although the circle of intersection with the caustic surface continues to contract. Beyond F the caustic surfaces are passed, and no part of the area is specially illuminated.

As a simple example of a symmetrical system let us take the case of parallel rays QR, 0A (fig. 7), incident upon a spherical mirror AR. By the law of reflexion the angle OR 7 = angle ORQ — angle çOR. Hence the triangle R7O is isosceles, and if we denote the radius of the surface OA. by r, and the angle AOR by α, we have <⅛ = 5~ l -

2 cos a

If F be the geometrical focus, OF = AF=⅜r. If a be a small angle, the longitudinal aberration Fg=Og ’- OF = ⅜r (sec α-l) = jα 2 r, in which AR=ra, Focal Lines.— In the general case of a small pencil of rays there is no one point which can be called the geometrical focus. Consider the corresponding small area of the orthogonal surface and its two sets of lines of curvature. Of all the rays which are contiguous to the central ray there are only two which intersect it, and these will in general intersect it at different points. These points may be regarded as foci, but it is in a less perfect sense than in the case of symmetrical pencils. Even if we limit ourselves to rays in one of the principal planes, the aberration is in general a quantity of the first order in the angle of the pencil, and not, as before, a quantity of the second order. If, however, we neglect this aberration and group the rays in succession according to the two sets of lines of curvature, we see that the pencil of rays passes through two focal lines perpendicular to one another and to the central ray, and situated at the centres of curvature of the orthogonal surface. At some intermediate place the section of the pencil is circular.

It happens not unfrequently that the pencil under consideration forms part of a symmetrical system, but is limited in such a manner that the central ray of the pencil does not coincide with the axis of the system. The plane of the meridian of the orthogonal surface is called the primary plane, and the corresponding focus, situated on the caustic surface, the primary focus. The secondary focus is on the axis of symmetry through which every ray passes. The distinction of primary and secondary is also employed when the system, though not of revolution, is symmetrical with respect to a plane passing through the central ray, this plane being considered primary.

The formation of focal lines is well shown experimentally by a plano-convex lens of plate-glass held at an obliquity of 20° or 30° in the path of the nearly parallel rays, which diverge from a small image of the sun formed by a lens of short focus. The convex face of the lens is to be turned towards the parallel rays, and a piece of red glass may be interposed to mitigate the effects of chromatic dispersion.

To find the position of the focal lines of a small pencil incident obliquely upon a plane refracting surface of index μ.

The complete system of rays issuing from Q (fig. 8).and refracted at the plane surface CA is symmetrical about the line QC drawn through Q perpendicularly to the surface. Hence, if QA be the central ray of the pencil, the secondary focus q i lies at the intersection of the refracted ray with the axis. If φ be the angle of incidence, φ' of refraction, AQ = w, Ag , 2 =¾ then

22, = sinø =/i · (1 )

u sin φ'

To find the position of the primary focus g 1 , let QA' be a neighbouring ray in the primary plane (that of the paper) with angles of incidence and refraction φ + δφ and φ' + δφ', Aq i — v 1 . We have

AA ' cos φ = uδφ, AA' cos φ'=v 1 δφ'; moreover, by the law of refraction, cos φ δφ—μ cos φ'δφ'; and thus

¾ 1 _ Μ cos 2 Φ' ( 2 ). u ~ cos 2 φ

If the refracting surface be curved, with curvature r -1 , we get by similar reasoning μ c os 2 <∕>' _ cos 2 <∕> _ μ cos φ' - cos φ . v 1 u ~ r μ 1 _ μ COS φ' - COS φ . v 2 u ~ r in which (1) and (2) are of course included as particular cases.

When the incidence is direct, cos Φ' = 1, cos φ = 1, and ¾ = υ 1 .

In this case (3) and (4) become

(5).

V u r To find the positions of the focal lines of a pencil refracted obliquely through a plate of thickness t and indece μ.

If φ be the angle of incidence (and emergence), φ' the angle of refraction of the ray QAST (fig. 9), S7 1 = v 1 , Sg , a =∙υ 2 , AQ=m, we get by successive applications of (1) and (2)

, ^cos 2 φ v 1 = u + s∈-,, (6); μ cos- φ v s =u + 7 , (7).

μ cos φ

If the incidence be direct, 0 1 =0 a =u4- A (8).

Thus, if we interpose a plate between the eye and an object, the effect is to bring the object apparently nearer by the amount (9) .

On this result is founded a method for determining the refractive index of materials in the form of plates. A set of cross wires is observed through a magnifying glass. On interposition of the plate the glass must be drawn back through a distance given by (9) in order to recover the focus. If we measure this distance and the thickness of the plate we are in a position to determine the refractive index.

Prism,. — By a prism is meant in optics a portion of transparent material limited by two plane faces which meet at a finite angle in a straight line called the edge of the prism. A section perpendicular to the edge is called a principal section.

Parallel rays, refracted successively at the two faces, emerge from the prism as a system of parallel rays. The angle through which the rays are bent is called the deviation.

The deviation depends upon the angles of incidence and emergence; but, since the course of a ray may always be reversed, the deviation is necessarily a symmetrical function of these angles. The deviation is consequently a maximum or a minimum when a ray within the prism is equally inclined to the two faces, in which case the angles of incidence and emergence are equal. It is in fact a minimum; and this position of the prism is described as the position of minimum deviation, and is usually adopted for the purposes of measurement.

The relation between the minimum deviation D, the angle of the prism i, and the refractive index μ is readily found. In fig. 10 the internal angles <∕>', ψ ’ are each equal to ∣ι. The external angles φ, ψ are also equal, and are connected with φ' by the law of refrac tion sin φ = μ sin φ'. The deviation is 2(≠ - φ'). Hence

_ sin ø _ sin ⅜(D + i) μ, ~ sin φ' ~ sin ⅛i ’ and this is the formula by which the refractive index is usually determined, since both D and i can be measured with great precision.

The instrument now usually employed for this purpose is called a goniometer or spectrometer. Parallel rays are provided by a collimator, consisting of an object-glass and telescope-tube, by means of which the subject of examination, either a fine slit or a set of cross wires, is seen as if it were at an infinite distance. The parallel rays from the collimator, after reflexion from a face or refraction through the body of the prism, are received by a telescope also provided with a set of cross wires at its focus. The table upon which the prism is supported, as well as the telescope, are capable of rotation about a vertical axis, and the position of either can be read off at any time by means of graduated circles and verniers.

As a preliminary to taking an observation it is necessary to focus the collimator and telescope. The first step is to adjust the eye-lens of the telescope until the cross wires are seen distinctly and without effort. The proper position depends, of course, upon the eyesight of the observer, and is variable within certain limits in virtue of the power of accommodation. It is usually best to draw out the lens nearly to the maximum distance consistent with distinct vision. The telescope is now turned to a distant object and focused by a common motion of the cross wires and eye-lens, until both the object and the cross wires are seen distinctly at the same time. The final test of the adjustment is the absence of a relative motion when the eye is moved sideways across the eye-piece. The collimator is now brought opposite to the telescope and adjusted until the cross wires in its focus behave precisely like the distant object.

To measure the angle of a prism it may be placed with its edge vertical upon the table, in a symmetrical position with respect to the collimator (fig. 11). The telescope is then successively brought into such positions that the cross wires of the telescope coincide with the cross wires of the collimator when seen by reflexion in the two faces. The difference of the readings is twice the angle of the prism.

Another method is also often employed in which the telescope is held fixed and the prism is rotated. The angle between the two positions of the table found by use in succession of the two faces is the supplement of the angle of the prism.

Suppose next that we wish to determine D for the given prism and for sodium light. The slit of the collimator is backed by a sodium flame, the telescope is adjusted for direct vision of the slit, and the reading taken. The prism is now placed upon the table, and rotated until the deviation of the light from its original direction when seen through the prism is a minimum. The difference of the readings for the two positions of the telescope is the value of D. The angle to be observed may be doubled by using the deviation in both directions. In this case no direct reading in the absence of the prism is required.

The following table of indices of refraction is taken from Watt’s Dictionary of Chemistry, article “Light.” [table]

A selection from some results given by Hopkinson, relating to Chance’s glasses, may be useful to those engaged in the designing of optical instruments. D is the more refrangible of the pair of sodium lines; b is the most refrangible of the group of magnesium lines; (G) is the hydrogen line near G. [table]

To determine the index of refraction of a liquid it must of course be placed in a hollow prism, whose faces are formed of some transparent material, usually of glass. The following results of Dale and Gladstone show the influence of temperature upon the refracting power of some important liquids. They relate to the soda flame, or the line D in the solar spectrum. [table]

Refractive Indices of Bisulphide of Carbon for the several Fixed, Lincs. [table]

The rapid alteration of refractive power with temperature is a serious obstacle to the use of bisulphide of carbon prisms for exact purposes. Not only does the dispersive power vary from day to day, but inequalities of temperature in the various parts of the liquid at any one moment disturb the optical uniformity, and are thus the cause of bad definition. A difference of l o Cent, alters the index about as much as a change in the light from one of the two D lines to the other, so that a variation of one degree within the prism may be expected to prevent the satisfactory resolution of this double line.

Excellent results have recently been obtained by Liveing with prisms containing aqueous solution of iodide of potassium and mercury. This liquid can be brought up to a density as high as three times that of water, and gives a powerful dispersion. Some difficulty has, however, been experienced in finding a suitable cement for the faces. Bisulphide of carbon prisms are usually cemented with a mixture of glue and treacle.

For many purposes the deviation of the light in passing through an ordinary prism is objectionable. In such cases recourse may be had to direct vision prisms (fig. 12), in which two materials, usually flint and crown, are so combined that the refractions are equal and opposite for a selected ray, while the dispersions are as unequal as may be. The direct vision prism may be contrasted with the achromatic lens (see Light). In the first the object is to obtain dispersion without refraction, and in the second to obtain refraction without dispersion.

Compound prisms, composed of a flint between two crowns, are also made, in which the action of the crown is not carried so far as to destroy the deviation due to the flint. By this construction a larger angle is admissible for the more dispersive material, but it is not clear that any sufficient advantage is gained.

The principle of the compound prism is carried to its limit by employing media of equal refracting power for the part of the spectrum under examination. For this purpose bisulphide of carbon and flint glass may be chosen. With Chance’s “dense flint” the refractions are the same, and the difference of dispersions is about as great as for “double-extra-dense flint” and crown. A dozen glass prisms of 90° may be cemented in a row on a strip of glass and immersed in a tube of bisulphide of carbon closed at the ends by glass plates. To vary the ray, which passes without deviation, ether may be mixed with the bisulphide.

The formation of a pure spectrum, which may be either thrown upon a screen or photographic plate, or received at once by the eye armed with a magnifier, has been explained under Light. It sometimes happens that the object is not to see the spectrum itself, but to arrange a field of view uniformly illuminated with approximately homogeneous light. For this purpose the pure spectrum is received upon a screen perforated by a narrow slit parallel to the fixed lines. The light which passes this second slit (eyeslit) is approximately homogeneous. Suppose that it corresponds to the red of the spectrum. The eye, placed immediately behind the eye-slit, receives only red light, and, if focused upon the prism, sees a red field of view whose brightness is uniform if the light falling in different directions upon the original slit be uniform. To secure the fulfilment of the last condition we may use the light from an overcast sky, or that of the sun reflected from a large surface of white paper. If it be desired to work by artificial light, an Argand gas flame diffused by an opal globe will be found suitable. When the adjustments are correct the tint should be perfectly uniform. Any difference of colour on the two sides of the field of view is an indication that the screen is not in its proper place.

The most important application of this arrangement is to the investigation of compound colours, as carried out by Maxwell. If light be admitted also through a second slit, displaced laterally from the position occupied by the first, a second spectrum overlapping the former will be thrown upon the screen, and a second kind of light will be admitted to the eye. In this way we may obtain a field of view lighted with a mixture of two or more spectrum colours, and we may control the relative proportions by varying the widths of the slits. For instance, by mixing almost any kind of red with any kind of green not inclining to blue we may match the brightest yellows, proving λvhat so many find it difficult to believe, that yellow is a compound colour. In Maxwell’s systematic examination of the spectrum, mixtures of three colours were used, and the proportions were adjusted so as to match the original white light incident upon the apparatus.

A similar arrangement (with one original slit) was employed by Helmholtz in his examination of a fundamental question raised by Brewster. The latter physicist maintained that there was abundant evidence to show that light of definite refrangibility was susceptible of further analysis by absorption, so that the colour of light (even of given brightness) could not be defined in terms of refrangibility or wave-length alone. The appearances which misled Brewster have since been explained as the effect of contrast or of insufficient purity. It is obvious that light, e.g., from the red end of the spectrum, may be contaminated with light from some other part, say the yellow, in such proportion that though originally entirely preponderant it may fall into the second place under the action of a medium very much more transparent to yellow than to red. To obtain light of sufficient purity for these experiments Helmholtz found it advisable to employ a double prismatic analysis. A spectrum is first thrown upon a screen perforated by a slit in the manner already described. The light which penetrates the second slit, already nearly pure, is caused to pass a second prism by the action of which any stray light is thrown aside. Using such doubly purified light, Helmholtz found the colour preserved, whatever absorbing agents were brought into play. Light of given refrangibility may produce a variety of effects, visual, thermal, or chemical, but (apart from polarization) it is not itself divisible into parts of different kinds. If yellow light produces the compound sensation of yellow, we are to seek the explanation in the constitution of the retina, and not in the divisibility of the light.

In all accurate work with the prism the use of a collimating lens to render the incident light parallel is a matter of necessity. If the incident rays diverge from a point at a finite distance, the pencil after emergence will be of a highly complicated character. There are, however, cases in which a collimator is dispensed with, and thus it is a problem of interest to find the foci of a thin pencil originally diverging from a point at a moderate distance. Even when a collimator is employed, the same problem presents itself whenever the focusing is imperfect. For the sake of simplicity the pencil is supposed to pass so near the edge of the prism that the length of path within the glass may be neglected in comparison with the distances of the foci.

We denote as usual the angles of incidence and emergence by Φ, ψ, and the corresponding angles within the glass by φ', φ'. The distance AQ from the edge of the prism to original source is denoted by u ; the corresponding distances for the primary and secondary f°ci ft, g ’ 2 by r υ v 2∙ By successive applications of the results already proved for a single refraction, we get cos 2 <⅛' cos 2 ≠

υ.,=u, υ 1 =—£ ~ i ιc fl); cos 2 <∕> cos 2 ψ v ' ’ so that υ 1 _ cos 2 ≠' cos-ψ _ (μ 2 - 1) tan 2 ≠ + μ 2 v 2 ~ cos 2 ≠ cos 2 ^' ~ (μ-'-l)tan 2 ^ + j u 2 '

In order that the primary and secondary foci may coincide we must have ψ=φ ; that is to say, the ray must pass with minimum deviation. This is sometimes given as a reason why this arrangement should be adopted in spectroscopes; but in reality, since the slit is parallel to the edge of the prism, a slight elongation in this direction of the image of a point is without detriment to the definition. Hence a good image will be seen when the telescope is adjusted for the primary focus; and it is not clear that any improvement λvould arise from coincidence of the two foci, the question being in fact one of aberration. The position of minimum deviation is, however, usually adopted for the sake of definiteness, and sometimes it is convenient that the fixed lines and the extremities of the slit (or the markings produced by dust) should be in focus together.

The deviation is a symmetrical function of φ and φ, and therefore is not altered by an interchange of these angles. The corresponding values of v are thus by (1) reciprocals, and their product is equal to w 2 . This principle has been ingeniously applied by Schuster to the adjustment for focus of the telescope and collimator of a spectroscope. The telescope is so placed that the deviation necessary to bring the object upon the cross wires is greater than the minimum, and the prism is adjusted in azimuth until the effect is produced, that position being chosen for which the angle of incidence is greater than the angle of emergence, so that r 1 is greater than u. After focusing the telescope the prism is turned into the other position which gives the same deviation, and the collimator is focused, the telescope remaining untouched. The prism is next brought back to the first position, and the telescope is again focused. A few repetitions of this operation, always focusing the telescope in the first position of the prism and the collimator in the second, will bring both into perfect adjustment for parallel rays.

Lenses.— The usual formula for the focal length of lenses (vol. xiv. p. 593),

)-⅛-υ(i r -i)

ignores the fact that the various parts of a lens bounded by spherical surfaces have not the same focus, and is applicable in strictness only when the aperture is small. It is not necessary here to repeat the process by which (1) is usually obtained, but before passing on to give the formulae for the aberration of lenses it may be well to exhibit the significance of (1) from the point of view of the wavetheory.

Taking the case of a convex lens of glass, let us suppose that parallel rays DA, EC, GB (fig. 13) fall upon the lens ACB, and are collected by it to a focus at F. The points D, E, G, equally distant from

ACB, lie upon a front of the wave before it impinges upon the lens. The focus is a point at which the different parts of the wave arrive at the same time, and that such a point can exist depends upon the fact that the propagation is slower in glass than in air. The ray ECF is retarded from having to pass through the thickness (i) of glass by the amount (μ - l)i. The ray DAF, which traverses only the extreme edge of the lens, is retarded merely on account of the crookedness of its path, and the amount of the retardation is measured by AF - CF. If F is a focus these retardations must be equal, or

AF-CF=(∕*-l)i,

Now if y be the semi-aperture AC of the lens, and ∕ be the focal length CF,

AF-CF = √{∕ 2 +y 2 } -∕=⅜ y-approximately,

. o∕2

whence f= —* 7 (2).

j 2(λt -l)i k '

In the case of plate-glass μ - 1 = ∣ nearly, and then the rule (2) may be thus stated: the semi-aperture is a mean proportional between the focal length and the thickness. The form (2) is in general the more significant, as well as the more practically useful, but we may of course express the thickness in terms of the curvatures and semi-aperture by means of In the preceding statement it has been supposed for simplicity that the lens comes to a sharp edge. If this be not the case we must take as the thickness of the lens the difference of the thicknesses at the centre and at the circumference. In this form the statement is applicable to concave lenses, and we see that the focal length is positive when the lens is thickest at the centre, but negative when the lens is thickest at the edge.

To determine practically the focal length of a convex lens we may proceed in several ways. A convenient plan is to set up a source of light Q (fig. 14) and a screen q at a distance exceeding four times the focal length, and to observe the two positions of the lens A, A' at which the source is in focus upon the screen. These positions are symmetrically situated, and the distance between them is observed. Thus

AQ = ⅜Qg ’ + ⅛AA', A 2 =^Q^-⅛AA'.

Now a j__i_

AQ + Aq~f ’ so that AQ. kq = Q<∕ 2 - AA' 2 f ~ AQ + A? -4 Q?

From the measured values of· Q⅛ and A A',/ can be deduced.

If A and A' coincide, the conjugate foci Q and q are as close as possible to one another, and then ∕ = ∣Q<∕.

The focal length on a concave lens may be found by combining it with a more powerful convex lens of known focus.

Aberration of Lenses.— The formula (1) determines the point at which a ray, originally parallel to the axis and at but a short distance from it, crosses the axis after passage through the lens. When, however, the ray considered is not quite close to the axis, the point thus determined varies with the distance y. In the case of a convex lens the ray DH (fig. 15), distant HC ( = y) from the axis, crosses it after refraction at a point F' which lies nearer to the lens than the point F determined by (1), and corresponding to an infinitely small value of y. The distance F'F is called the longitudinal aberration of the ray, and may be denoted by 8f.

The calculation of the longitudinal aberration as dependent upon the refractive index (μ) and the anterior and posterior radii of the surfaces (r, s') is straight forward, but is scarcely of sufficient interest to be given at length in a work like the present. It is found that

- i ∕^=⅛{⅜ + (ψ-l)(l-j∏∕3...(3X

r, s, and f being related as usual by (1).

The first question which suggests itself is whether it is possible so to proportion r and s that the aberration may vanish. Writing for brevity R, S, F respectively for r -1 . s -1 , ∕ -1 , and taking ∕∕P ’

G = so that - S = (G/μ) - R, we get μ- 1

⅛ + (i y 2 - τ)(y - ⅛) 2 = G i W+ 2 ) - RG(2μ+l)+μG 2 } -r∕ H J-ΟΊ P (2μ + l)G V, μ 3 F 3 4μ-1.

-Gj√(μ + 2).R- 2 ^ + 2 J + (/t _ 1)84(M + 2 )···(4 )·

Since μ >1, both terms are of the same sign; and thus it appears that the aberration can never vanish, whatever may be the ratio of r to s. Under these circumstances all that we can do is to ascertain for what form of lens the aberration is a minimum, the focal length and aperture being given. For this purpose we must suppose that the first term of (4) vanishes, which gives

2(μ + 2)(μ-l)

r ~ μ(2μ + l)

The corresponding value of-s is

2(μ+2)(μ-l) s --ΓΓμ- 2μ 2 f (6)> so that - szr - m( 2 ^+1) m

4+μ-2μ 2 '*

In the case of plate-glass μ=1 · 5 nearly, and then from (5), (6), (7) r =⅛ ^ s =⅛∕> -s: r=6:l.

Both surfaces are therefore convex, but the curvature of the anterior surface (that directed towards the incident parallel rays) is six times the curvature of the posterior surface. By (3) the outstanding aberration is

15 y 2 δ ∙^ - 14 ∕ < 8 ) ,

The use of a plano-convex lens instead of that above determined does not entail much increase of aberration. Putting in (3) s= αo, and therefore by (1) τ=⅛f t we get

*∕=4τ (9) ·

This is on the supposition that the curved side faces the parallel rays. If the lens be turned round so as to present the plane face to the incident light we have r=∞, -s=⅛f, and then s -f=-lτ— (10).

nearly four times as great.

For a somewhat higher value of μ the plano-convex becomes the form of minimum aberration. If s=∞ in (6), 4+μ-2μ 2 =0, whence μ=l'69.

If μ be very great, we see from (5) and (6) that r and s tend to become identical with f.

For the general value of μ the minimum aberration corresponding to (7) is by (4)

_ . f _ X*μ-i) ∏ 1 )

δf ∙f 8(μ-l) 2 (μ + 2)

The right-hand member of (11) tends to diminish as μ increases, but it remains considerable for all natural substances. If μ=2, f f 16

Oblique Pencils.— Hitherto we have supposed that the axis of the pencil coincides with the axis of the lens. If the axis of the pencil, though incident obliquely, pass through the centre of the lens, it suffers no deviation, the surfaces being parallel at the points of incidence and emergence. In this case the primary and secondary foci are formed at distances from the centre of the lens which can only differ from the distance corresponding to a direct pencil by quantities of the second order in the obliquity. Hence, if the obliquity be moderate, we may use the same formulae for oblique as for direct pencils.

The consideration of excentrical pencils leads to calculations of great complexity, upon which we do not enter.

Chromatic Aberration.— The operation of simple lenses is much interfered with by the variation of the refractive index with the colour of the light. The focal length is decidedly less for blue than for red light, and thus in the ordinary case of white light it is impossible to obtain a perfect image, however completely the spherical aberration may be corrected. From the formula for the focal length we see that f- μ ∖ r s ) μ-l f' so that

-v=⅛∕≡

or the longitudinal chromatic aberration varies as the focal length and as the dispersive power of the material composing the lens. The best image will be formed at a position midway between the two foci, and the diameter d of the circle over which the rays are spread bears the same ratio to the semi-aperture of the lens (y) that δ∕ bears to f. Hence δμ d= — i y.

M-l

The diameter of the circle of chromatic aberration is thus proportional to the aperture and independent of the focal length ; and, since the linear dimensions of the image are proportional to the focal length, the confusion due to chromatic aberration may be considered to be inversely as the focal length. Before the invention of the achromatic object-glass this source of imperfect definition was by far the most important, and, in order to mitigate its influence, telescopes were made of gigantic length. Even at the present day the images of large so-called achromatic glasses are sensibly impaired by secondary chromatic aberration, the effect of which is also directly as the aperture and inversely as the focal length.

Achromatic Object-glasses.— It has been shown in vol. xiv. p. 595 that the condition of achromatism for two thin lenses placed close together is

⅛⅛7=<> (1).

∕* -1 ∕ Μ -1 ∕

in which f f' are the focal lengths of the two lenses, and δ∕z∕(∕χ - 1), δμ∕(μ' - 1) the dispersive powers of the two kinds of glass. In practice crown and flint glass are used, the dispersive power of the flint being greater than that of the crown. Thus f' is negative and numerically greater than f, so that the combination consists of a convex lens of crown and a concave lens of flint, the converging power of the crown overpowering the diverging power of the flint. When the focal length F of the combination is given, the focal lengths of the individual lenses are determined by (1) in conjunction with

1 1 1

f=7 + 7 γ (2) ·

The matter, however, is not quite so simple as the above account of it might lead us to suppose. In consequence of what is called the irrationality of spectra, the ratio of dispersive powers of two media is dependent upon the parts of the spectrum which we take into consideration. Whatever two rays of the spectrum we like to select, we can secure that the compound lens shall have the same focal length for these rays, but we shall then find that for other rays the focal length is slightly different. In the case of a single lens the focal length continually diminishes as we pass up the spectrum from red to violet. By the use of two lenses the spectrum, formed as it were along the axis, is doubled upon itself. The focal length is least for a certain ray, which may be selected at pleasure. Thus in the ordinary achromatic lens, intended for use with the eye, the focal length is a minimum for the green, and increases as we pass away from the green, whether towards red or towards blue. Stokes has shown that the secondary colour gives a sharp test of the success of the achromatizing process.

“The secondary tints in an objective are readily shown by directing the telescope to a vertical line separating light from dark, such as the edge of a chimney seen in the shade against the sky, and covering half the object-glass with a screen having a vertical edge. So delicate is this test that, on testing different telescopes by well-known opticians, a difference in the mode of achromatism may be detected. The best results are said to be obtained when the secondary green is intermediate between green and yellow. This corresponds to making the focal length a minimum for the brightest part of the spectrum.

“To enable me to form a judgment as to the sharpness of the test furnished by the tint of the secondary green, as compared with the .performance of an object-glass, I tried the following experiment. A set of parallel lines of increasing fineness was ruled with ink on a sheet of white paper, and a broader black object was laid upon it as well, parallel to the lines. The paper was placed, with the black lines vertical, at a considerable distance on a lawn, and was viewed through two opposed prisms, one of crown glass and the other of flint, of such angles as nearly to achromatize each other in the positions of minimum deviation, and then through a small telescope The achromatism is now effected, and varied in character, by moving one of the prisms slightly in azimuth, and after each alteration the telescope was focused afresh to get the sharpest vision that could be had. I found that the azimuth of the prism was fixed within decidedly narrower limits by the condition that the secondary green should be of such or such a tint, even though no attempt was made to determine the tint otherwise than by memory, than by the condition that the vision of the fine lines should be as sharp as possible. Now a small element of a double object-glass may be regarded, so far as chromatic compensation is concerned, as a pair of opposed prisms; and therefore we may infer that the tint of the secondary green ought to be at the very least as sharp a test of the goodness of the chromatic compensation as the actual performance of the telescope.”

In the case of photographic lenses the conditions of the problem are materially different. It is usually considered to be important to secure “coincidence of the visual and chemical foci,” so that the sensitive plate may occupy the exact position previously found by the eye for the ground glass screen. For this purpose the ray of minimum focus must be chosen further up in the spectrum. If, however, the object be to obtain the sharpest possible photographs, coincidence of visual and chemical foci must be sacrificed, the proper position for the sensitive plate being found by trial. The middle of the chemically-acting part of the spectrum, which will vary somewhat according to the photographic process employed, should then be chosen for minimum focus.

When the focal lengths of the component lenses have been chosen, it still remains to decide upon the curvatures of the individual faces. Between the four curvatures we have at present only two relations, and thus two more can be satisfied. One of these is given by the condition that the first term in the expression for the aberration—that proportional to the square of the aperture—shall vanish for parallel rays. As to the fourth condition, various proposals have been made. If equal and opposite curvatures are given to the second and third surfaces, the glasses may be cemented together, by which some saving of light is effected. Herschel proposed to make the aberration vanish for nearly parallel, as well as for absolutely parallel, rays. This leads to a construction nearly agreeing with that adopted by Fraunhofer.

The following results are given by Herschel for the radii of the four surfaces, corresponding to various dispersive powers, and to mean refractive indices l - 524 (crown) and 1 ’ 585 (flint). The focal length of the combination is taken equal to 10, and, as well as the radii, is measured in arbitrary units; so that all the numbers in the table (with the exception of the first column) may be changed in any proportion. [table]

The general character of the combination is shown in fig. 16.

The radii of the first and fourth surfaces within practical limits are so nearly constant that Herschel lays down the following rule as in all probability sufficiently exact for use. A double object-glass will be free from aberration, provided the radius of the exterior surface of the crown lens be 6’720 and of the flint 14’20, the focal length of the combin ation being 10’000, and the radii of the interior surface being computed from these data, by the formulae given in all elementary works on optics, so as to make the focal lengths of the two glasses in the direct ratio of their dispersive powers.

Numerous experiments have been made with the view of abolishing the secondary spectrum. Theoretically, if three different kinds of glasses are combined it will generally be possible to make the focal lengths of the combination equal for any three selected rays of the spectrum. Or the ingredients of one of the glasses may be mixed in such proportions as to suit the requirements of the problem when combined with crown. In this way Stokes has succeeded in constructing a small object-glass free from secondary colour, but it is doubtful whether the practical difficulties could be overcome in the construction of a large object-glass, where alone the outstanding chromatic aberration is important.

The practical optician is not limited to spherical surfaces, and the final adjustment of the aberration of large objectglasses is controlled by the action of the polishing tool. It is understood that some of the best makers apply a local correction, according to the methods developed by Foucault for mirrors. The light from a natural or artificial star is allowed to fall upon the lens. At the focus is placed a small screen, which is gradually advanced so as to cut off the light. The eye is immediately behind the screen and is focused upon the lens. If there are no imperfections the illumination falls off very suddenly, the surface of the mirror passing from light to dark through a nearly uniform grey tint. If, however, from uniform aberration, or from local defects, any of the light goes a little astray, the corresponding parts of the surface will show irregularities of illumination during the passage of the screen, and in this manner a guide is afforded for the completion of the figuring.

Töppler has developed the idea of Foucault into a general method for rendering visible very small optical differences. Instead of a mere point of light, it is advisable to use as source an aperture (backed by a bright flame) of sensible size, and bounded on one side by a straight edge. An image of this source is formed at a considerable distance by a lens of large aperture and free from imperfections, and in the plane of the image is arranged a screen whose edge is parallel to the straight edge of the image, and can be advanced gradually so as to coincide with it. Behind this screen comes a small telescope through which the observer examines the object placed near the lens. When the light is just cut off by the advancing screen, the apparatus is in the most sensitive state, and the slightest disturbance of the course of the rays is rendered evident. To show the delicacy of the arrangement Töppler introduced into the cone of light a small trough with parallel glass sides containing distilled water. A syphon dipped under the surface and discharged distilled water from another vessel, and it was found almost impossible so to control the temperatures that the issuing jet should remain invisible. Not only were sound-waves in air, generated by electric sparks, rendered visible, but their behaviour when reflected from neighbouring obstacles was beautifully exhibited.

An apparatus on this principle may often be employed with advantage in physical demonstrations,—for instance, for the exhibition of the changes of density in the neighbourhood of the electrodes of a metallic solution undergoing electrolysis. The smallest irregularity that could be rendered visible would be such as would retard transmitted light by a moderate fraction of the wave-length.

In objectives for photographic use the requirements are in many respects different from those most important in the case of telescopes. A flat field, a wide angle of view— in some cases as much as 90° —freedom from distortion, and a great concentration of light are more important than a high degree of definition. As a rule, photographs are not subjected to the ordeal of a high magnifying power. Usually the picture includes objects at various distances from the camera, which cannot all be in focus at once. That the objects at one particular distance should be depicted with especial sharpness would often be rather a disadvantage than otherwise. A moderate amount of “diffusion of focus” is thus desirable, and implies residual aberration. In some lenses an adjustment is provided by means of which the diffusion of focus may be varied according to the circumstances of the case.

For landscapes and general purposes a so-called single lens is usually employed. This, however, for the sake of achromatism, is compounded of a flint and a crown cemented together; or sometimes three component lenses are used, the flint being encased in two crowns, one on each side. To get tolerable definition and flatness of field a stop must be added, whose proper place is some little distance in front of the lens.

For portraiture, especially before the introduction of the modern rapid dry plates, a brilliant image was a necessity. This implies a high ratio of aperture to focal length, which cannot be attained satisfactorily with any ft>rm of single lens. To meet the demand, Petzval designed the “portrait-lens,” in which two achromatic lenses, placed at a certain distance apart, combine to form the image. This construction is so successful that the focal length is often no more than three times the available aperture. When stops are employed to increase the sharpness and depth of focus they are placed between the lenses.

Vision through a Single Lens.— A single lens may be used to improve the vision of a defective eye, or as a magnifying glass. A normal eye is capable of focusing upon objects at any distance greater than about 8 inches. The eyes of a short-sighted person are optically too powerful, and cannot be focused upon an object at a moderate distance. The remedy is of course to be found in concave glasses. On the other hand, persons beyond middle life usually lose the power of seeing near objects distinctly, and require convex glasses.

A not uncommon defect, distinct from mere short or weak sight, is that known as astigmatism. In such cases the focal length varies in different planes, and at no distance is the definition perfect. Many people, whose sight would not usually be considered inferior, are affected by astigmatism to a certain extent. If a set of parallel black lines ruled upon white paper be turned gradually round in its own plane, it will often be seen more distinctly and with greater contrast of the white and black parts in one azimuth than in another. When the focal line on the retina is parallel to the length of the bar, the definition (as in the case of the spectroscope) is not much prejudiced, but it is otherwise when the bars are turned through a right angle so as to be perpendicular to the focal line.

In extreme cases a remedy may be applied in the form of glasses of different curvatures in perpendicular planes, so adjusted both in form and position as to compensate the corresponding differences in the lens of the eye.

The use of a lens as a magnifier has been explained under Microscope. The simplest view of the matter is that the lens, consistently with good focusing, allows of a nearer approach, and therefore of a higher visual angle, than would otherwise be possible.

Telescope, <L ’ c.— In a large class of optical instruments an image of the original object is first formed, and this image is examined through a magnifier. If we use a single lens merely for the latter purpose, the field of view is very restricted. A great improvement in this respect may be effected by the introduction of a field-∖ens. The ideal position for the field-lens is at the focal plane of the object-glass. The image is then entirely uninfluenced, and the only effect is to bend round the rays from the margin of the field which would otherwise escape, and to make them reach the eye-lens, and ultimately the eye. If the field-lens and the eye-lens have nearly the same focal length an image of the object-glass will be formed upon the eye-lens, and through this small image will pass every ray admitted by the object-glass and field-lens.

However, to obtain a sufficient augmentation of the field of view it is not necessary to give the field-lens the exact position above mentioned, and other considerations favour a certain displacement. For example, it is not desirable that dust and flaws on the field-lens should be seen in focus. In Huygens’s eye-piece the field-lens is displaced from its ideal position towards the object-glass. In Ramsden’s eye-piece, on the other hand, the focal plane of the object-glass is outside the system. This eye-piece has the important advantage that cross wires can be placed so as to coincide with the image as formed by the object-glass. The component lenses of a Ramsden’s eye-piece are sometimes achromatic. For further particulars, with diagrams, on the subject of eye-pieces, see Microscope.

In large telescopes the object-glass is often replaced by a mirror, which may be of speculum metal, or of glass coated chemically with a very thin layer of polished silver. The mirror presents the advantage (especially important for photographic applications) of absolute achromatism. On the other hand, more light is lost in the reflexion than in thé passage through a good object-glass, and the surface of the mirror needs occasional re-polishing or re-coating. For fuller information see Telescope.

The function of a telescope is to increase the “apparent magnitude” of distant objects; it does not increase the ‘apparent brightness.” If we put out of account the loss of light by reflexion at glass surfaces (or by imperfect reflexion at metallic surfaces) and by absorption, and suppose that the magnifying power does not exceed the ratio of the aperture of the object-glass to that of the pupil, under which condition the pupil will be filled with light, we may say that the “apparent brightness” is absolutely unchanged by the use of a telescope. In this statement, however, two reservations must be admitted. If the object under examination, like a fixed star, have no sensible apparent magnitude, the conception of “apparent brightness” is altogether inapplicable, and we are concerned only with the total quantity of light reaching the eye. Again, it is found that the visibility of an object seen against a black background depends not only upon the “apparent brightness” but also upon the apparent magnitude. If two or three crosses of different sizes be cut out of the same piece of white paper, and be erected against a black background on the further side of a nearly dark room, the smaller ones become invisible in a light still sufficient to show the larger. Under these circumstances a suitable telescope may of course bring also the smaller objects into view. The explanation is probably to be sought in imperfect action of the lens of the eye when the pupil is dilated to the utmost. The author of this article has found that in a nearly dark room he becomes distinctly short-sighted, a defect of which there is no trace whatever in a moderate light. If this view be correct, the brightness of the image on the retina is really less in the case of a small than in the case of a large object, although the so-called apparent brightnesses may be the same. However this may be, the utility of a night-glass is beyond dispute.

The general law that (apart from the accidental losses mentioned above) the “apparent brightness” depends only upon the area of the pupil filled with light, though often ill understood, has been established for a long time, as the following quotation from Smith’s Optics (Cambridge, 1738), p. 113, will show.

“Since the magnitude of the pupil is subject to be varied by various degrees of light, let NO be its semi-diameter when the object PL is viewed by the naked eye from the distance OP; and upon a plane that touches the eye at 0, let OK be the semi-diameter of the greatest area, visible through all the glasses to another eye at P, to be found as PL was; or, which is the same thing, let OK be the semi-diameter of the greatest area inlightened by a pencil of rays flowing from P through all the glasses; and when this area is not less than the area of the pupil, the point P will appear just as bright through all the glasses as it would do if they were removed; but if the inlightened area be less than the area of the pupil, the point P will appear less bright through the glasses than if they were removed in the same proportion as the inlightened area is less than the pupil. And these proportions of apparent brightness would be accurate if all the incident rays were transmitted through the glasses to the eye, or if only an insensible part of them were stopt.”

Resolving Power of Optical Instruments.— According to the principles of common optics, there is no limit to the resolving power of an instrument. If the aberrations of a microscope were perfectly compensated it might reveal to us worlds within a space of a millionth of an inch. In like manner a telescope might resolve double stars of any degree of closeness. The magnifying power may be exalted at pleasure by increase of focal length and of the power of eye-pieces; and there are at any rate some objects, such as the sun, in dealing with which the accompanying loss of light would be an advantage rather than the contrary. How is it, then, that the power of the microscope is subject to an absolute limit, and that if we wish to observe minute detail on the over-lighted disk of the sun we must employ a telescope of large aperture? The answer requires us to go behind the approximate doctrine of rays, on which common optics is built, and to take into consideration the finite character of the wave-length of light.

A calculation based upon the principles of the wavetheory shows that, no matter how perfect an object-glass may be, the image of a star is represented, not by a mathematical point, but by a disk of finite size surrounded by a system of alternately dark and bright rings. Airy found that if the angular radius of the central disk (as seen from the centre of the object-glass) be θ, 2R the aperture, λ the wave-length, then i = l∙21W⅛, showing that the definition, as thus limited by the finiteness of λ, increases with the aperture.

In estimating theoretically the resolving power of a telescope on a double star we have to consider the illumination of the field due to the superposition of the two independent images. If the angular interval between the components of the double star were equal to 2θ, the central disks would be just in contact. Under these conditions there can be no doubt that the star would appear to be fairly resolved, since the brightness of the external ring systems is too small to produce any material confusion, unless indeed the components are of very unequal magnitude. The diminution of star disks with increasing aperture was observed by W. Herschel; and in 1823 Fraunhofer formulated the law of inverse proportionality. In investigations extending over a long series of years, the advantage of a large aperture in separating the components of close double stars was fully examined by Dawes.

The resolving power of telescopes was investigated also by Foucault, who employed a scale of equal bright and dark alternate parts; it was found to be proportional to the aperture and independent of the focal length. In telescopes of the best construction the performance is not sensibly prejudiced by outstanding aberration, and the limit imposed by the finiteness of the waves of light is practically reached. Verdet has compared Foucault’s results with theory, and has drawn the conclusion that the radius of the visible part of the image of a luminous point was nearly equal to half the radius of the first dark ring.

The theory of resolving power is rather simpler when the aperture is rectangular instead of circular, and when the subject of examination consists of two or more light or dark lines parallel to one of the sides of the aperture. Supposing this side to be vertical, we may say that the definition, or resolving power, is independent of the vertical aperture, and that a double line will be about on the point of resolution when its components subtend an angle equal to that subtended by the wave-length of light at a distance equal to the horizontal aperture.

The resolving power of a telescope with a circular or rectangular aperture is easily investigated experimentally. The best object is a grating of fine wires, about fifty to the inch, backed by a soda-flame. The object-glass is provided with diaphragms pierced with round holes or slits. One of these, of width equal, say, to one-tenth of an inch, is inserted in front of the object-glass, and the telescope, carefully focused all the while, is drawn gradually back from the grating until the lines are no longer seen. From a measurement of the maximum distance the least angle between consecutive lines consistent with resolution may be deduced, and a comparison made with the rule stated above.

Merely to show the dependence of resolving power on aperture it is not necessary to use a telescope at all. It is sufficient to look at wire-gauze backed by the sky, or by a flame, through a piece of blackened cardboard pierced by a needle and held close to the eye. By varying the distance the point is easily found at which resolution ceases; and the observation is as sharp as with a telescope. The function of the telescope is in fact to allow the use of a wider, and therefore more easily measurable, aperture. An interesting modification of the experiment may be made by using light of various wave-lengths.

In the case of the microscope the wave-theory shows that there must be an absolute limit to resolving power independent of the construction of the instrument. No optical contrivances can decide whether light comes from one point or from another if the distance between them do not exceed a small fraction of the wave-length. This idea, which appears to have been familiar to Fraunhofer, has recently been expanded by Abbe and Helmholtz into a systematic theory of the microscopic limit. See Microscope.

Similar principles may be applied to investigate the resolving power of spectroscopes, whether dispersing or diffracting. Consider for simplicity any combination of prisms, anyhow disposed, but consisting of one kind of glass. Let a be the final width and μ the index of a parallel beam passing through, and let the thicknesses of glass traversed by the extreme rays on either side be t 2 and t 1 . It is not difficult to see that, if the index be changed to μ + δμ, the rays will be turned through an angle θ given by a

Now, if the two kinds of light correspond to a double line which the instrument can just resolve, we have θ= ∖∣ a, and thus i 2 -i 1 =λ∕δμ, a formula of capital importance in the theory of the dispersing spectroscope. In a well-constructed instrument, t v the smaller thickness traversed, may be small or negligible, and then we may state the law in the following form :—the smallest thickness of prisms necessary for the resolution of a double line whose indices are μ and μ + δμ is found by dividing the wave-length by δμ.

As an example, let it be required to find the smallest thickness of a prism of Chance’s “extra dense flint,” necessary for resolution of the soda-lines.

By Cauchy’s formula for the relation between μ and λ we have μ=A+Bλ~ 2 , δμ=- 2Bλ" 3 δλ,

From the results given by Hopkinson for this kind of glass we find

B=∙984×10- 1 °, the unit of length being the centimetre. For the two soda-lines λ=5∙889 × 10~ 5 , δλ= ∙006 × 10~ 5 ; ajrd thus the thickness t necessary to resolve the lines is λ 4 10 10 λ 4 i =2Ba=Fs∞X= 1 ' 02 ce " tlmetra ∙

the meaning of which is that the soda-lines.will be resolved if, and will not be resolved unless, the difference of thicknesses of glass traversed by the two sides of the beam amount to one centimetre. In the most favourable arrangement the centimetre is the length of the base of the prism. It is to be understood, of course, that the magnifying power applied is sufficient to narrow the beam ultimately to the diameter of the pupil of the eye; otherwise the full width would not be utilized.

The theory of the resolving power of a diffracting spectroscope, or grating, is even simpler. Whatever may be the position of the grating, a double line of wave-lengths λ and λ-t-δλ will be just resolved provided

δλ = J_

Λ mn where n is the total number of lines in the grating, and m is the order of the spectrum under examination.

If a grating giving a spectrum of the first order and a prism of extra dense glass have equal power in the region of the soda-lines, the former must have about as many thousand lines as the latter has centimetres of available thickness.

The dispersion produced by a grating situated in a given manner is readily inferred from the resolving power. If a be the width of the beam after leaving the grating, the angle δθ, corresponding to the limit of resolution, is λ∕α, and thus δθ _mn

δλ - a *

Thus the dispersion depends only upon the order of the spectrum, the total number of lines, and the width of the emergent beam.

An obvious inference from the necessary imperfection of optical images is the uselessness of attempting anything like an absolute destruction of aberration. In an instrument free from aberration the waves arrive at the focal point in the same phase. It will suffice for practical purposes if the error of phase nowhere exceeds ⅛λ. This corresponds to an error of ∣λ in a reflecting and ⅜λ in a (glass) refracting surface, the incidence in both cases being perpendicular.

If we inquire what is the greatest admissible longitudinal aberration in an object-glass according to the above rule, we find δ∕=λα^ 2 ,

α being the angular semi-aperture.

In the case of a single lens of glass with the most favourable curvatures, δf is about equal to ∕α 2 ; so that α 4 must not exceed λ∕∕. For a lens of 3-feet focus this condition is satisfied if the aperture do not exceed 2 inches.

When parallel rays fall directly upon a spherical mirror the longitudinal aberration is only about one-eighth as great as for the most favourable-shaped single lens of equal focal length and aperture. Hence a spherical mirror of 3-feet focus might have an aperture of 2⅜ inches, and the image would not suffer materially from aberration.

On general optics the treatises most accessible to the English reader are Parkinson’s Optics (3d ed., 1870) and Glazebrook’s Physical Optics (1883). Verdet’s Leçons d’optique physique is an excellent work. Every student should read the earlier parts of Newton’s Optics, in which are described the fundamental experiments upon the decomposition of white light ( r. )