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tion of the mirrors: that is, OKB, or OKb=2 LKM. For OKB-OKa+aKB=OKa+OKA=aKA=2LKM

=OKA+AKb=OKA+OKa=2LKM.

Upon this principle is the common reflecting quadrant for measuring angles constructed, so that the angular distance between any object and its image formed by two reflexions, is always double to the inclination of the mirrors. This may also be elegantly demonstrated, in another way, independent of what has been already proved, in the following manner.

Let DCK* represent the quadrant, with a movable index CX, which carries with it the central speculum C, while the small speculum F remains fixed on the radius of the instrument, and parallel to the central speculum, when the index coincides with CD, the other radius; the beginning of the graduations on the arch being at D. Let HQ and FQ be perpendicular to these specula respectively, so that the angle HQF will show their inclination equal to the angle DCX. Now let a ray of light from the object at S fall upon the mirror at C, and be reflected to F, and from thence be again reflected to the eye at I. The image of the object S will therefore be seen in the line IFO, parallel to HCQ. Now SC being produced to I, the angular distance SIO of the object and its image=SCH= HCF CFI=2QFI=2FQH-twice the angle of inclination between the mirrors. So that by the motion of the index over any arch, the image of the object moves at the same time over twice the angular distance. Hence the arch of the instrument is divided into half degrees, which are counted as whole degrees, to give the angle at sight.

In every position of the instrument, in the plane of

[graphic]

the objects, whose angular distance is to be measured, the angle at I remains the same, and consequently the ́reflected image will be stationary, which makes the instrument of such inestimable value at sea. For although the motion of the ship will cause a motion in the image reflected from the central mirror; yet as the second mirror fixed on the radius of the instrument, which reflects the image to the eye, will communicate to it an equal motion in an opposite direction, it will, by that means, destroy the effect of the first motion, and conse. quently the image will appear to be quiescent: so that the motion of the vessel, so fatal to observations taken with other instruments, will not in the least affect such as are taken with this. There are many advantages attending the use of this instrument above any others, besides that already mentioned; such as, that it will give the altitude of the sun or of a star, or the distance of the moon from them, generally, to less than a single minute; and that, with respect to the sun, when it does not even shine so bright as to project a shadow.

OF THE REFRACTION OF RAYS

FROM SPHERICAL SURFACES.

THE fundamental law, that the angles of incidence and reflexion, are equal, obtains in spherical surfaces, whether concave or convex, as well as in plane mirrors.

Parallel rays, that are incident near the vertex, are reflected by a concave mirror, converging to a focus, in the axis, half way between the vertex and center of concavity; and by a convex mirror, diverging, as if they came from a point in the axis, at the same distance from the center or vertex. Because the angle of incidence is equal to the angle of reflexion, ABC-CBF BCF.* Therefore CF-FB-FV, when B and V coin

* See Plate 13, Fig. 1.

cide. The point F is therefore called the principal focus, or focus of parallel rays.

Hence if rays diverge from the principal focus of a concave mirror, or converge to the principal focus of a convex mirror, they will be reflected parallel to the axis. If they diverge from the center of a concave mirror, or converge to the center of a convex mirror, they will be reflected back in the same line. In a concave mirror, if they diverge from a point between the focus and center, they will be reflected converging to the axis beyond the center, and if they diverge from a point beyond the center, they will be reflected back converging to a point in the axis between the center and principal focus. But if they diverge from a point between the vertex and principal focus, they will be reflected back diverging from the axis, and vice versa. Converging rays are reflected converging to a point in the axis between the vertex and principal focus. What is here said of diverging rays falling upon a concave mirror, is also true of converging rays falling on a convex mirror, and vice versa.-All evident from the equality of the angles of incidence and reflexion.

Diverging rays falling near to the vertex of a spherical mirror, if they diverge from a point more distant from the vertex than the principal focus, will be collected into a focus, or diverge as if they came from a focus, at such a distance from the principal focus, that this distance will be a third proportional to the distance of the radiant point from the principal focus, and the principal focal distance. That is, if RB* be a diverging ray falling on the spherical surface at B near the axis CV, and reflected back to f in the concave mirror, or as if it came from f in the convex mirror, then it will be RF: FC:: FC: Ff.

* See Plate 13, Fig. 2 and 3.

Let CD, CE, be drawn parallel to the incident and reflected rays, respectively, and cutting them in the points D and E. Then CEBD will be a parallelogram, of which all the sides in its evanescent state, as B approaches to V, will be equal to each other and to half the diagonal or to FC. Then because the triangles REC, RBf are similar, it will be as RE=RF: EC=FC:: RB=RV: Bf=Vf.

Cor. As RF: RFFC=RV :: FC=VF: FC+Ff= Vf. That is, in words at length: If rays diverge from, or converge to, a radiant point, and fall upon a concave mirror, they will be reflected to a focus in the axis, which may be found by the following proportion.

As the difference or sum, of the distance of the radiant point from the vertex, and principal focal distance, is to the distance of the radiant point, so is the focal distance of parallel rays, to the focal distance of the diverging or converging rays from the vertex. If D=the distance of the radiant point, F=the focal distance of parallel rays, and f=the focal distance of the diverging or converging DF rays, then

D+F

=f; or, D+F: D:: F: f.

In the first term of the above proportion, for diverging rays falling on a concave mirror, or for converging rays falling upon a convex mirror, use the difference between the distance of the radiant point and principal focal distance; and use their sum for converging rays falling upon a concave mirror, and also for diverging rays falling on a convex mirror.

Hence we see the reason why a convex mirror always has a positive focus; because the first term of the above proportion is always a positive quantity, excepting when both the first and second terms, (D=the distance of the

radiant point being negative) are negative, in which the fourth term must also be positive.

We see also the reason why a concave mirror may sometimes have a negative, and sometimes a positive, focus. When D, or the distance of the radiant point, is positive and less than the principal focal distance, then the focus of diverging rays will be negative.

As any ray passing through the center of a mirror is reflected back in the same line, as being perpendicular to the surface, and any other ray is reflected so as to cross the perpendicular in the focus, whether that ray diverged from or converged to any point in the perpendicular; and as a ray may be drawn through the center from any point of an object, and other rays diverging from the same point or converging to it, will be reflected to some point as their focus in the said perpendicular; the image of any point of an object will be found in the perpendicular from that point. Hence it is easy to understand, that if the object be so small or so far distant from the reflecting surface or center, that all its points are nearly at equal distances from it, all the corresponding points of the image will be nearly at other equal distances from the said surface or center. As the object and image are both terminated by the perpendiculars, which cross each other in the center, they must be both seen under the same angle from the center, and therefore their magnitudes must be in the same ratio to their distances from the center. They are also seen under the same angle from the vertex of the mirror, and consequently their magnitudes are also in the ratio of their distances from the vertex.

In all spherical mirrors, the image is erect, when it is on the same side of the center with the object, but inverted when they are on different sides. Because the

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