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...