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image being always in the perpendicular drawn through the correspondent point of the object to the reflecting surface, and the perpendiculars crossing each other only in the center; it must follow, that if both the object and image be on the same side of the center, the image will be erect, but if on different sides, inverted.

Hence we have an easy way of determining the place, position, and magnitude of an image formed by any spherical surface, by drawing lines from the extremities of the object both through the center and the vertex, and producing them both ways, and their intersections will geometrically determine the place and magnitude of the image.

In a concave mirror, as the object and image are viewed under the same angle at the center and vertex, if the eye be close to the mirror, the apparent diameters of the object and image will be the same. If the object touch a concave mirror, the image will also touch it, and both will be equal in magnitude. If the object be nearer to the mirror than its principal focus, but does not touch it, the image is erect, distinct and behind the mirror, greater and more distant than the object: but decreases as the eye departs.

If the object be in the principal focus of a concave mirror, the image is at an infinite distance behind the mirror, magnified, erect, and distinct. As the rays arc in this case reflected parallel, the place of the image will be at an infinite distance, and on account of its remoteness it will appear magnified, and as the perpendiculars do not intersect each other, it will be erect, and because of the parallelism of the rays, it will be seen as distinctly as distant objects are usually seen: and in this case, the apparent magnitude of the image is not altered by the motion of the eye.

If the object be farther from the vertex of a concave mirror than the principal focus, and the eye nearer than the place of the distinct image, the image will be behind the mirror, erect, magnified, and confused, but if the eye be more remote than the place of the distinct picture, the image is before the mirror, distinct and inverted; and is increased in magnitude as the eye departs, provided that it does not depart beyond the place of the distinct image.

When the image appears before a concave mirror, if the object be nearer to the mirror than the center, the image will be larger, if more remote, less, and if in the center, equal to the object; and if the eye move one way, the image moves the contrary way; but if the mirror move, the image moves the same way.

If the eye be in the center of a concave mirror, it can see nothing but its own image, because no rays can come to it but what diverged from itself; its image must therefore appear diffused over the whole mirror, and be infinitely magnified and confused.

When the eye is in the principal focus of a concave mirror, the apparent magnitude of an image is not altered by the departure of the object; but when the eye is nearer to the mirror, the image decreases with the removal of the object, but increases, when the eye is more distant, provided it be not removed so far as to bring the image between the eye and the mirror.

The image of an object seen by a convex mirror is always behind the mirror, erect, distinct, less than the object, and nearer to the mirror than the object; and when either the eye or object recedes from the mirror, the apparent diameter decreases, and is never equal to the object but when it touches the mirror. So that the object and image approach towards, and recede from,

the mirror at the same time. If either the object or mirror move, the image moves the same way: but if the eye move, the image seems to move the contrary

way.

From repeated experiments it is found that the best composition for the most copious reflexion of light is made in the following manner. Let the whole mass be divided into fifty equal parts, then 32 must be of copper, 15 of tin, and of silver, brass, and arsenic, each one part. This metal, when broken, should appear of a bright, glassy, and quick-silver complexion. But if it appear of a dead white, more tin must be added: so that 32 ounces of copper will require 16 ounces of tin, if it be very pure. But if the metal when broken appear bluish and rough, more copper or brass must be added.

CAUSTICS BY REFLEXION.

WHAT We have hitherto said concerning the focus of rays reflected from a spherical surface, only respects the rays that are incident near to the axis of the mirror; because those rays that are reflected from more distant parts are collected into different points in the axis, nearer to the vertex than the principal focus. The intersections of these reflected rays, with each other, before they reach the axis, form luminous curves which are denominated caustics, which are convex towards the axis, and have their cusps in the principal focus. They are easily seen on the surface of milk or any white fluid in a tea-cup, when the light of the candle is reflected from the opposite side. The more distant that any ray is from the axis, the nearer to the vertex will its reflected part intersect the axis, and consequently as a ray nearer to the axis is reflected to a more distant point, it must intersect the more distant ray, when re

flected from the spherical surface. In like manner, the rays that are incident on the mirror, still nearer to the vertex, cross each other, still farther from the mirror and nearer to the axis, in different points, until they finally concur in the principal focus. All these points of intersection of contiguous rays form the caustic curves.

As the reflected parts of any two contiguous rays, on each side of the axis, must intersect each other at different distances both from the axis and vertex of the mirror, there will be two caustics formed at the same time, one on each side of the axis, whose cuspides unite in the principal focus.

REFRACTION OF LIGHT

THROUGH PLANE AND SPHERICAL SURFACES.

WHEN the rays of light fall perpendicularly on any medium, they suffer no change in their direction, but only in their velocity, as we have demonstrated before; but when they fall obliquely on any surface, they will be refracted towards or from the perpendicular, according as the medium, into which they enter, is denser or rarer than that through which they moved before their incidence upon it.

When parallel rays fall obliquely on a plane surface of a medium of different density, they will be parallel after refraction; for, as they have the same inclination to the surface, they will all suffer an equal degree of refraction, being rays of the same kind.

If they diverge before their incidence on the plane surface of a denser medium, they will be made to diverge less; and if they converge, to converge less: but when they are incident upon the plane surface of a rarer medium, diverging rays are made to diverge more, and converging rays to converge more; because, in the first

case, they are refracted towards, and in the second case, from the perpendiculars.

Parallel rays, entering a denser medium through a convex surface, are made to converge; as the perpendiculars, to which they are refracted, meet in the center of convexity. But if they enter through a concave surface, they are made to diverge; because the perpendiculars, to which they are refracted, diverge in the den. ser medium from a point in the rarer medium. But being refracted the contrary way or from the perpendiculars, when they enter a rarer medium, either through a convex or concave surface of the rarer, they are made to diverge or converge according as the surface of the rarer medium is convex or concave.

Diverging rays, entering into a denser medium through a convex surface of the denser medium, or a concave surface of the rarer, are made to converge, be parallel, or to diverge less, according to the degree of their divergency. If they diverge from the focal distance of parallel rays when they enter the convex surface of the denser medium, they will proceed parallel after refraction, the medium having the same refractive power, which way soever the rays proceed; and if they diverge from a point nearer to the denser medium, they will proceed diverging but in a less degree; but if they diverge from a more distant point, they will be made to converge. If they diverge from the center of concavity of the rarer medium, they fall in with the perpendiculars and suffer no refraction; if from a point nearer to the rarer medium, being refracted from the perpendicular, they are made to diverge more; but if from a point more distant than the center of concavity, they are made to diverge less, to be parallel, or to converge. If they diverge from a point, which would be the focal

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