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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. '.
cos. A' cos. A
Hence, and by the question,
= max. or min.
3 cos.. 0 6. cos. O'sin.
sin. 0 =
Hence, by substitution, we get, after the proper reductions,
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.
824. If A and A' denote the distances of the foci of incident and refracted rays, we have (Coddington, p. 56)
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
and this substituted in the expression for the maximum gives
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
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,)
Hence the distances of the object O and image O' by reflection from the centre are
Again, since the distance of the image by reflection from the reflecting surface is
the distance of its image caused by refraction from that surface is got from the expression
Hence the distance of this image, which is concentric and similar to the former, from the centre is
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)
Also, since the distance of the principal focus of the sphere from the centre is (Coddington, p. 66,)
:. the visual angle of the object when at the principal focus is
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,)
✓ (4co—4) and the equation between its rectangular co-ordinates is known, vize
62 y? =
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)