LIGHT.

395

Xight to the caustic, which continue to the surface C, and join Q C. Then it is obvious, that any small pencil Q C, Q Cdiverging from Q, will form a focus at q (Art. 134, &c.) from which it will afterwards diverge, and fall on the eyeat E, nearly as if the rays came from a mathematical point; and from what was said in Art. 161 and 162, itappears that the density of rays in the cone gE is infinitely greater than in any adjacent cone having the eye forits base; so that q will appear as an image of Q, more or less confused, in proportion to the degree of curvatureof the caustic at q ; for it is evident, that if the curvature be great, the assumed concentration of any small finitepencil Q C C' in one mathematical point q, will deviate more from truth than if the caustic approach nearly to astraight line.

Corol. As the eye shifts its place, the apparent position of an object seen in a reflecting or refracting surfaceshifts also, for as E varies, the tangent E q shifts its place on- the caustic, and the point of contact q, or theplace of the image shifts.

This doctrine may be illustrated by a very familiar instance. If we look through a surface of still water, notvery deep, but having a level horizontal bottom, the bottom will not appear a plane, but will seem to rise on allsides, and approach nearer the surface the more obliquely we look. To explain this, let Q be a point in thebottom, and let QPe be the course of the pencil of rays by which an eye at e sees it (fig. 39) on the visual ray.The point in the caustic to which e P produced is a tangent, is Y; and from the form of the caustic DYll (seeArt. 238) it is obvious, that Y is nearer the surface the more oblique e P is to it. The apparent figure of thebottom will therefore be thus determined. From the eye E (fig. 69) draw any line E g to the point G of thesurface; and having drawn P Y parallel to E G, touching the branch DYB of the caustic having Q, verticallybelow E for a radiant point in Y, prolong E G to H, making G H = P Y, then will H be the image of the pointQ' in the bottom, belonging to the caustic D' II B'; and the locus of II, or the apparent form of the bottom,will be the curve D F II, having a basin-shaped curvature at D, a point of contrary flexure at F, and anasymptote C G K coinciding with the surface.

But, to return to the case of images formed by rays incident at very small obliquities and nearly central,the following rules for determining their places, magnitudes, and apparent situations in all cases of sphericalsurfaces, will be convenient to bear in memory, and will need no express demonstration to the reader of the fore-going pages.

Rule 1. Any image formed, or about to be formed, by converging rays, or from which rays diverge, may beregarded as an object.

Rule 2. In spherical reflectors the object and its image lie on the same side of the principal focus. They movein contrary directions, ami meet at the centre and surface of the reflector. The distance of the image from tileprincipal focus anti centre M liad by the pfOpOftlOIl

Q F : F E :: E F : F (jr: : Q E : E 9,

and the image is erect when the object and surface lie on the same side of the principal focus ; but inverted whenon contrary sides. The relative magnitudes of the object and image (being as their distances from the centre)are given by the proportion

object : image : : Q F : F E : : distance of the object from the principal focus ; focal length of reflector.

Rule 3. In thin lenses, of all species, if Q be the place of the object, q of its image, E the centre of the lens,F the principal focus of rays incident in a contrary direction , then will the object and image lie on the same, oropposite side of the lens, according as the object and lens lie on the same or opposite sides of the principalfocus F. In the former case the image is erect, in the latter inverted, with respect to the object. The distanceof the image from the lens, or from the object, is had by the proportions

QF:FE::QE:E(7; QF:QE::QE:Q9 ;and the magnitude of the object is to that of the image as the distance of the object from F is to the focal length,or as Q F ; F E.

Part I.

342.

343.

Apparentfigure of thehorizontalbottom ofstill water.Hg. 39.

Fig. 69.

344.

Rules forfinding theplace, &c.of an image.

345.

346.

Rule foirr Rectors.Fig. 16

347.

Rule forlenses.

Ride 4. In all combinations of reflectors and lenses, the image formed by one is to be regarded as the object, 34 .whose image is to be formed by the next, and so on, till we come to the last.

It has been already remarked (Art. 6) that visible objects are distinguished from optical images by this, that 349.from the former light emanates in all directions, whereas in the latter it emanates only in certain directions.

This is an important limitation in practical optics. A real object can be seen whenever nothing opaque isinterposed between it and the eye. An image can only be seen when the eye is placed in the pencil of rayswhich goes to form it, or diverges from it. Thus in the case represented in fig. 62, except the eye be placedsomewhere in the space D q p H, it will see no part of the image, B q D and Ap II being the extreme raysrefracted by the lens from the extremities of the object.

The brightness of an image is, of course, proportional to the quantity of light which is concentrated in each Brightnesspoint of it; and, therefore, supposing no aberration, as the apparent magnitude of the lens or mirror which forms of images.

dl*6cl of* ot)l6Ct

it, seen from the object x ---. Or, since the area of the object : that of the image : : (distance) 1

area of image

of object from lens : (distance) 2 of image ; and since the apparent magnitude of the lens seen from the object

( diameter A 0 . , , . , .

distance from —If—t / ’ l ^ e brightness or degree of dlummation of the image is as the apparent

3 F 2

is as i