REFLECTION FROM CONCAVE MIRRORS.

301

concave mirror, of which c is the geometrical centre, and parallel rays, as defgh, be incident

F

B

B

-d

-e

-f

9

-h

upon it, they will be reflected according to the general law of reflection (431), and consequently be made to converge towards a point F, situate midway between

the centre c and the point E, F being consequently equal to half the radius of the concavity of the mirror.

It is obvious that all the luminous undulations producing the rays defgh will be reflected towards F, and, arriving there. at the same instant, will cause any particles of ether there situated to be acted upon and agitated with an intensity corresponding to the united force (422) of all the undulations. propagated from the reflecting surface; on which account all the light and heat belonging to the incident rays will become concentrated at F, and luminous and calorific effects of corresponding intensity will be excited on any body placed at that spot. This point is hence termed the focus or fire-place of the mirror AEB for parallel rays; the distance FE being termed the principal focal distance, or length of the mirror.

437. If diverging rays be incident upon a concave mirror, they will be conveyed to a focus which differs from the point F in the last figure, approaching the centre of the mirror's

concavity. Thus, if rays diverging from aluminous source, as a lighted candle P, be incident upon a concave mirror, they will be reflected, accord

B

ing to the general law, to a

focus f much nearer c than the point r, or focus for parallel rays (436). If then the candle P be placed at ƒ, the luminous rays will be reflected by the mirror to a focus at P; hence P and ƒ are termed conjugate foci, for either becomes the focus to a radiant point placed at the other. Whereas, in the case of parallel rays (436), if the source of light be

placed at F, the rays will be reflected in a parallel direction, and never meet at a focus. If the candle or other radiant point be placed nearer the mirror than its principal focus, its rays will be reflected, not parallel but divergent, as though they were evolved from some point placed behind the mirror. The conjugate focal distance for diverging rays may be in which d corresponds to the

found by the formula

d+r 2d xr

distance of the source of light from the mirror, and r to the radius of curvature of the latter.

438. When converging rays are incident on a concave mirror, they will be reflected to a focus further from the centre of the mirror's concavity than the principal focus or F (436), the reverse consequently of diverging rays. These rays, falling on a mirror, appear to converge towards a point situated behind it, and their focus may be found by the following formula, in which e corresponds to the distance of the point of convergence from the mirror, d and r retaining their former

439. When luminous rays are incident upon convex mirrors, they are acted upon in a manner opposite to that which they were by concave reflecting surfaces; for whilst a concave reflector lessens the divergency, and increases the convergency of all incident rays, a convex one increases their divergency and diminishes their convergency. Thus, if parallel

rays abcde be incident on the convex mirror AB, of which c is the centre of convexity, they will be reflected, according to

CAUSTICS BY REFLECTION.

303

the general law (431), in the direction a'b'd'e', as if they proceeded from a point r placed behind the mirror, which thus becomes the virtual, apparent, or negative focus of the

reflected rays.

The focal distance F for parallel rays is one half of the radius of the convexity of the mirror, and always situated behind the mirror, whilst in concave reflectors it is before it (436). In the case of diverging rays, the focal distance will be less, and for converging beams greater than y.

440. When luminous rays are incident upon a curved reflector, any point of its surface may be considered as an infinitely minute plane mirror (432), reflecting all the rays falling upon them. When a series of rays fall upon a surface thus constituted, they after reflection mutually intersect, and these points of intersection constitute a curved line, termed a caustic. To exhibit this caustic curve by reflection, nearly fill a glass tumbler with milk, or fit a circular piece of card into it about half an inch from the top, and, exposing the concavity of the glass to the sun or a candle, a brilliant double curve will be represented on the surface of the milk, or piece of paper.

441. Images are formed by spherical mirrors in the same manner as by plane ones, and differ from those produced by the latter instruments in being of a different size from the object.

Thus, if rays be supposed to emanate from a distant body, they will, on being incident on the concave mirror AB, of which c is the centre of concavity, be reflected to a focus at r, a little beyond the principal

focus (436), and there paint an image of the object ED, diminished in size, and, from the altered relative position of the rays after reflection (433), inverted in direction. The image

F will be extremely vivid from its being virtually illuminated by all the luminous rays incident on the mirror. The magnitude of the image r will be found to bear the same relation to ED as the distance of F from the mirror does to that of the object from it. If an object be placed at F, its image will be painted on a screen placed at ED, diffused over a large space, and consequently magnified.

H

442. In the case of convex mirrors, the images are in an erect position, much diminished in size, and behind the reflecting surface as in the plane B mirrors (432); for if an object DE be placed before a convex mirror AB, whose negative focus is at F, the luminous rays will, after incidence on AB, be reflected diverging; and being seen by a spectator at н, they will appear

to him as proceeding from an object de behind the mirror, and considerably smaller than DE, of which it is merely a diminished image.

CHAPTER XXI.

UNPOLARIZED LIGHT. (DIOPTRICS.)

Re

Law of Sines, 443. Refraction from dense through rare Media, 444. Index of Refraction, 445. Refraction through two Media, 446. Limit to Refraction, internal Reflection, 448-9. Refraction through Parallel Surfaces, 450-through Prisms, 451. Lenses, 452. fraction through Spheres, 453-4-through Convex Lenses, 455. Formula for Focal Lengths, 456. Refraction through Concave Lenses, 457-through Menisci and Concavo-convex Lenses, 459. Caustics by Refraction, 460. Formation of Images by Lenses, 461-2. Magnifying Power of Convex Lenses, 463-4. Spherical Aberration in Lenses, 465—in Mirrors, 466.

443. So long as a ray of light traverses an uniform medium, it continues its path in a right line, which it also preserves when it is incident on a diaphanous substance, in a direction perpendicular to its surface. But if it be incident in an oblique direction, it becomes somewhat bent, or refracted, out of its original course: this bending, or refraction, not being constant for every substance, as the direction of reflection (431) is, but varies considerably in different forms of matter. Thus, let AB be the surface of a refracting medium, as water; draw CD perpendicular to it, and let rɛ be a ray incident on AB at E, a certain portion will be reflected (431*), the remainder entering the medium, and instead of following the direction EG, will be refracted or bent towards D in the direction EK. The line FE will, therefore, represent the incident, and EK the refracted ray; FEC will be the angle of incidence, and DEK the angle of refrac