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distance of parallel rays entering the convex surface of the denser medium, they will be parallel after refraction in the rarer medium, and if from a point more or less distant, than the said focal distance, they will converge or diverge accordingly.

If diverging rays fall upon a concave surface of the denser medium, and diverge from the center of con. cavity, they fall in with the perpendiculars and suffer no refraction; but if they diverge from a point nearer to the surface, they are made to diverge less, being refracted towards the perpendiculars; and if they diverge from a point more distant than the center of concavity, they are made to diverge more. If diverging rays fall on the convex surface of a rarer medium, they are made to diverge more, being refracted from the perpendiculars.

If converging rays fall upon a convex surface of a denser medium, and if they converge to the center of convexity, they fall in with the perpendiculars and suffer no refraction: if they converge to a point nearer to the surface, they are made less converging, and if they converge to a point more distant than the center, they are made more converging. If converging rays fall upon the convex surface of a rarer medium, and if they converge to the center, they also fall in with the perpendiculars and suffer no refraction. But if they converge to a point nearer to the surface, they are rendered more converging, and if to a point more distant than the center, they are rendered less converging. The reason, in both cases, is, because the refracted rays are bent towards the perpendiculars in the denser medium, and from them in the rarer.

If converging rays fall upon a concave surface of the denser medium, they may be made to be less con

verging, parallel, or even diverging. But converging rays falling on the concave surface of a rarer medium are made more converging.

From what has been said, it is easy to understand the progress of the rays of light, after passing through lenses of every kind; whose density is always supposed to be different from that of the medium with which they are surrounded. And we may deduce this observation, in general, from what is before laid down; that when diverging rays are made to converge, by passing through a lens, the nearer the radiant point is to the lens, the farther is the focus from it, and vice versa; and when rays diverge from a point as far distant as the focus of parallel rays, they proceed parallel, as the lens has the same refractive power, which way soever the rays proceed. And if rays diverge after refraction, their focus is imaginary, and on the same side with the radiant point.

There is a certain point belonging to every lens, through which if a ray pass, it will proceed parallel to the direction it had before its incidence on the lens. This is called the center of the lens. It is any where in a transparent medium bounded by parallel surfaces; it is in the vertex of the single convex or concave lenses, within the double convex and concave lenses, and removed a little out of the meniscus, and nearer to the surface that has the greater curvity.

That a ray, which passes through the center of a lens, or during its passage through a meniscus tends to its center, is refracted in a direction parallel to that in which it was incident is thus demonstrated.

Let R, r,* be the centers of curvity of the sides of

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the lens, from which draw two parallel radii to the incident and emergent rays, and join the centers by the line Rr, intersecting the passing ray, produced, if necessary, in the point E. Now E is the center of the lens. And because the triangles REA, rEB are similar, the angle RAE=rBE, RE, rE, will be in the constant ratio of the radii, so that the point E will be constant in the same lens; and as the incident and emergent rays have the same inclination to the perpendiculars, they must be equally refracted in contrary directions at each surface of the lens, and therefore they must be parallel to each other.

THE FOCUS OF RAYS BY REFRACTION.

THE focus of parallel rays, refracted through a plano-convex or plano-concave lens, if they fall near to the vertex, is at the distance of twice the radius of the sphere, to which the lens is ground: which may

thus be demonstrated.

It was before observed, that when the refraction is made out of glass into air, the sine of the angle of incidence is to that of refraction as 2 : 3. Now let C* be the center of convexity or concavity of the lens, and CF the axis of the lens, parallel to which let the ray AB be incident on the lens near to the vertex V, and the refracted ray will be in the direction BF, and F will be the focus, either real or imaginary, and VF will be 2CV. For since DBE-ABC-BCF=the angle of incidence, is to DBF, the angle of refraction, nearly as their sines, in such small angles, that is, as 2:3, their difference EBF-BFC=1, will be to the angle of incidence BCF as 1:2, and their opposite sides CB, and BF are in the same proportion, as, 1:2,

* Plate 13, fig. 7. and Plate 14, fig. 1.

that is as CV: VF, when the point B approaches near to V; therefore VF-2CV.

Hence, if the lens were equally convex or concave on both sides, the focal distance of parallel rays would be only one half of what is above determined, or at the distance of the radius of the sphere to which the lens was ground. Because the rays would now suffer an equal degree of refraction at each side, or twice as much as if it were plane on one side. This is called the principal focus of parallel rays.

Hence, also, if rays diverge from the principal focus of parallel rays, they will be refracted parallel to the axis of the convex lens. The same also will be the direction of the refracted ray, after passing through a concave lens, when the incident ray converges to the focus of parallel rays. Because the lens has the same refractive power, which way soever the incident ray

comes.

If rays diverge from a radiant point and be refracted by a convex lens, they will converge to a point in the axis more distant than the focus of parallel rays. The distance of this point from the lens, called the focus of diverging rays, is found by saying, as the difference between the distance of the radiant point and focal distance of parallel rays is to the distance of the radiant point, so is the focal distance of parallel rays to the focal distance of diverging rays. But if the rays converge to a radiant point, the first term of the proportion is the sum of the distance of the radiant point, and focal distance of parallel rays. That is if D=the distance of radiant, F-focal distance of parallel rays, and f the focal distance of diverging or converging rays. D+F:D :: F: f

Let AC* be a lens, whose focal distance for parallel rays is CF, let this distance be set on both sides of the lens, and perpendiculars be erected from it to intersect the incident and emergent rays, and from the points of intersection, let lines be drawn through the center of the lens, which will be respectively parallel to the incident and refracted rays. Because if two rays, GC, GA diverge from the focus of parallel rays, and one of them GC pass through the center of the lens, the other Af will be refracted parallel to it, the first suffering no refraction. Hence the triangles RGC, RAf, will be similar, and therefore RG: GC :: RA : Af, that is, for diverging rays passing through a convex lens, D-F: F:: D: f, and for converging rays, D+ F: F:: D:f. The imaginary focal distance of converging rays passing through a concave lens is the same with that of diverging rays passing through a convex lens, viz. D—F: F : : D : f. and of diverging rays passing through a concave lens, the same with that of converging rays passing through a convex lens, viz. D+F: F:: D:f

Hence, since the focus of parallel rays, refracted through a plano-convex lens or a plano-concave, is at the distance of twice the radius of the sphere=2r; this may be substituted in the above formulas instead of F. And since F-r in double convex or concave lenses, of equal curvity on both sides, the above theorems will 2 Dr Dr become D+2r

f. and D+r

=f, for single or double

lenses of equal curvity on both sides, respectively; the lower signs belonging to convex lenses, and diverging rays, or converging rays and concave lenses; and the upper signs for converging rays and convex lenses, or diverging rays and concave lenses.

* Plate 14, fig. 2, 3, 4, 5.

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