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822. If p be the number of reflections, A the 2 of incidence, and B that of refraction; then the between the incident and emergent ray is (Wood, Prop. XCIII.)

20 - 2 (p + 1) ('. Also for the rays to emerge parallel, we must have (Wood, Prop. XCIV.)

tan. 8 : tan, a' :: p +1:1 and sin. 0 = m sin. 6 = sin. '.

3

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cos. A' cos. A

Hence, and by the question,

8

4
G

= max. or min.

3
and putting its differential = 0, we get

do
4 cos. A (cos. 6'

oʻsin. 0')
do

3 cos.. 0 6. cos. O'sin.

But since

sin. 0 =

sin. 6

3

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Hence, by substitution, we get, after the proper reductions,

3 cos.20
sin. 8 sin. A

cos. A

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823. Letr be the radius of the hemisphere, x and y the co-ordinates, measured from the edge of the bowl, of the nearest point which comes into view when it is filled with water; then it readily appears that the depth (d) of the image of this point : ::. : r.

.: dec. Hence

d : y :: sin, ' : sin. 0 :: 3 : 4 0 and e' being the angles of incidence and refraction.

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824. If A and A' denote the distances of the foci of incident and refracted rays, we have (Coddington, p. 56)

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mr A i. A' - a

- A = max.

(m - n) A + nr and putting its differential = 0, we get

(m n) mr a

-1=0 (m - n) A + nr

(m-n. A tnr) which gives

mr

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✓mt in

and this substituted in the expression for the maximum gives
A'- A =
Im - sn

Xr.
Nm trn

825. Since the object is small it may be considered a circular arc concentric with the sphere, and its image after reflection will therefore be a circular arc, also concentric with the sphere, and similar to the object, (Wood, Prop. XV.)

Again, if A denote the distance of the image from the refractor, in order that its image, caused by refraction, may be distinct, we must have (Wood, Prop. XXXIX.)

A - 9:r:: sin. I : sin. R :: 4 : 3

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7

Hence the distance of the image by reflection from the reflecting surface is

1/ and it is :. exterior to the sphere; and the distance of the object from the same surface is :: (Wood, p. 26,)

2r

T

3

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Hence the distances of the object O and image O' by reflection from the centre are

andr +

3r

r

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10

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Again, since the distance of the image by reflection from the reflecting surface is

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the distance of its image caused by refraction from that surface is got from the expression

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Hence the distance of this image, which is concentric and similar to the former, from the centre is

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a

Now let a be the distance of the eye from the centre; then its

26 distance from O" is

5 and the visual angle is

5 x 52 x 0 7x (26r-a)

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Also, since the distance of the principal focus of the sphere from the centre is (Coddington, p. 66,)

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:. the visual angle of the object when at the principal focus is

O

a + 2r

:: 0 : 0 :: a + 2r : 250 x 7 · (26r - a).

826. From the centre of the reflector let fall a I upon the rectilinear object produced; then taking this as an axis, the image will be a portion of a conic section, whose major-axis coincides with that I, and whose focus is the centre of the reflector. Let o be the traced angle, and p the radius vector of this conic section, c the distance of the foot of the object from the centre, and, the radius of the reflector ; then (Coddington p. 36,)

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e2 =

(4co—4) and the equation between its rectangular co-ordinates is known, vize

62 y? =

(a2 xP).

Again, let « be the inclination of the object to the given axis, d its length, and n the part of it produced. Also let o denote the inclination of the tangent at any point of the image to its major axis ; then

dy tan. Q = dx

su — x2 b (1? y)

y

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a

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