OPTICS is that branch of physical science which explains the formation of images, as depending on the known laws by which the modifications of light are governed. [Liana.] These images are formed either by polished reflecting surfaces or by transparent refract ing media. In the former case, the angles of incidence and reflection are equal ; in the latter, the sines of the angles of incidence and refraction are in a constant ratio for one and the same medium. The position and magnitude of the image of an object is easily ascertained, when we have previously ascertained the position of the image of a point, in reference to the position of the point itself and of the reflecting or refracting instrument ; in other words, when we have found the relation between the conjugate foci, so called because it universally holds in optics, that whichever focus be considered the object, the other will be the image. The principal focus of an instru ment is that to or from which a pencil of parallel rays falling perpen dicularly (or nearly so) on the instrument is made to converge or diverge after reflection or refraction. In a plane mirror the conjugate foci are similarly situated at opposite sides of the mirror [Limn]; consequently in this instance the instrument has no principal focus. Generally, the distance of the principal focus from the instrument is called the focal length of that instrument, whether a reflector or a refractor. Sind: conjugate foci are mutually such, it follows that rays proceeding from the principal focus will, after reflection or refraction, emerge in a parallel pencil. We shall now proceed to the relations existing between tho conjugate foci of spherical reflectors, observing that the axis of the instrument is the right line containing the centre of the spherical surface and the conjugate foci. The rays under consideration are them) which are directed nearly along the axis, and which therefore fall exceedingly nearly perpendicularly on the reflector.
Let nn it represent the section of a spherical reflector made by a plane passing through its axis, C its centre, A the focus of incident rays, A n an incident ray, A D C is the angle of incidence. Make the angle a D C = A n c, then a n c la the angle of reflection, and if the point of incidence n were infinitely near to the point n in the axis, then all the reflected rays of which the incidence was nearly perpendicular would converge to a, the latter would then be the locus conjugate to A, for if rays diverged from a they would after reflection evidently converge to A.
Now, if a straight line as c n bisect an angle of a triangle, as the angle A 'a a, it will divide the base into segments A c, c a proportional to the adjacent sides A D, n a (' Enc.,' book vi.), that is, Ac ca : D: Da ; but when D is infinitely near we may write An and na instead of AD and De, in which case we should have Ac:ca::Ania a. Let A D = a n = A', and the radius cri .a-r; then A c= A-r ; c a= r- A' ; whence A -r : r -A' : : A : A', or A' (A— r)= (r- A); therefore 2 A = r (A + Al, which may be also written in the form = We should have precisely the same investigation if we had supposed rays ass n to fall on the convex side converging to a focus A ; but being reflected in the direction De, they would appear to diverge from the conjugate focus a : hence the above formula applies to two eases, namely, when diverging rays fall on the concave surface, or converging rays on the convex surface, of a spherical reflector.
Example 1.-A candle is placed before a concave speculum at a distance of 3 feet from it : what will be the distance of its image from the same, the radius of the speculum being 2 feet / Here we have given A=3 feet, r = 2 feet, and to find A' we substitute 1 1 2 these numbers in the general formula - + =-, which thus becomes A r 2 1 ; whence -= - and therefore A'= -= foot; the image 3 A' A' 3 2 will consequently be 1 foot 6 inches in front of the speculum.