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788. Since the image is always at the same distance from the reflector as the object, the reflector will always bisect that part of the revolving line which is intercepted by the given fixed lines. Hence, if r, gu denote the distances of the axis of revolution from the two given lines, measured along the revolving line, when in such a position as to be equally inclined to both of them ; 28 be the inclination of the given lines, and 0 the angle described from t, and the radius vector of the required locus, it may easily be shewn that

cos.ß

go cos. (B+0) +r' cos. (B—) Х

cos.(B+0).cos. (B0) cos.B

{r sec. (B – ) + q sec. (B + 6)}

2

or

2

which is the polar equation to the required locus.

The student may amuse himself with deducing other equations to the curve, and by tracing it through all its ramifications.

789. Since, by hypothesis, the object is very distant compared with its magnitude, it may be considered a circular arc concentric with the reflector. Hence also the image is a circular arc similar and concentric with the object, (Wood, p. 113,) and if 0, o be the absolute magnitudes of the object and image, we have

0:0 :: 9:0 ' q and q' being the radii of the object and image.

Again, let the given distance between the eye and image bed, and a, a' the known at the eye subtended by the object and image, and x the distance of the object from the eye, and the radius of the reflector; then from the great distance of the object from the eye it may be considered a circular arc, having the eye for its centre,

..0 = X X a

O' = d x a'
: 3 x 4 : 4 x 4 :: q : 4
But q: 4 :: QF : FE :: f - q:f

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f-9 mr-m-1.9

fr (1 – m) ing

r- -(m-1) f Hence x xa: d xa :: f-9:f

:: rm - (m-1) f : q- m-if

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m#1.(2-x)

.:. r = 2.

m+1

m-1 which determines the magnitude of the reflector.

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and x-q is therefore known, which, being the distance of the centre of the reflector from the eye, gives the position of the reflector.

790. Let a be the altitude of the cylinder, r the radius of its base, also let m be the ratio of refraction ; then the depth of the image of the centre of the base is

a

m

and if o denote the inclination of the line joining the eye and edge of the cylinder to the vertical, we have

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which gives the direction of vision. Again, let h be the height of the

eye above the fluid, and x the distance of the person from the vessel ; then

mhr x = h tan. 0 =

a

791. The rays of light proceeding from the wick form a frustum of a cone, whose axis is parallel to the wall, and smaller end the top of the cover. This code of light being of indefinite length is cut by the plane of the wall, which, being parallel to the axis, the section will be the common hyperbola.

If the top of the cover be not horizontal, the luminous figure on the wall will be an ellipse.

792. Changing the notation to that of the problem, and making r positive and r negative, we have (Coddington, p. 65)

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the terms involving t, 6, &c. being neglected because of their comparative smallness.

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794. He must use a double concave lens, whose focal length is (Wood, p. 133,) QE X Eq 14 x 3

= 3 feet 9

inches. Qq

11

F=

795. Let o be the 2 of incidence of the given ray, r the radius of the reflector, and x the distance of the reflected ray

from the centre; then

d:r:: sin. Q: sin. (60 -0) and x ::: sin. Q: sin. (60 + 9). From the first proportion we get

dp 3
tan. Q =

2r + d
and this being substituted in the second, gives

rd

rtà

OPTICS.

Hence, since the geometrical focus bisects the radius, the distance of the reflected ray from it is

rd F

rud according as the focus of incident rays is nearer to the reflector than its centre is, or not.

796. Generally (Coddington, p. 66,)

2

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798. The density of the rays in the sun's image oc as the area of the aperture directly, and inversely as the focal length of the lens. If, therefore, d, d' be the densities for the sphere of water and the plano-convex lens of glass, we have

2
d : ď::

:
f f

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m =

and go' = 2r

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