Sidebilder
PDF
ePub

OPTICS.

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

202 (p+1) '.

Also for the rays to emerge parallel, we must have (Wood, Prop. XCIV.)

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

and putting its differential = 0, we get

4 cos. (cos. ' — ' sin. 0')

[ocr errors]

=

[merged small][merged small][ocr errors][ocr errors]

cos. ' sin.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small]

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

[merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small]

823. Let r 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;

[blocks in formation]

dy: sin. ' ; sin. 0 :: 3 : 4

and 'being the angles of incidence and refraction.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small]

824.

If A and A' denote the distances of the foci of in

cident and refracted rays, we have (Coddington, p. 56)

[ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

and this substituted in the expression for the maximum gives

[blocks in formation]

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 ▲ 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.)

Arr: sin. I sin. R :: 4 : 3

[blocks in formation]

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

[merged small][merged small][merged small][ocr errors][merged small]

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

[ocr errors][merged small][merged small][merged small][merged small][merged small]

Hence the distances of the object O and image O' by reflection

[merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

Again, since the distance of the image by reflection from the

reflecting surface is

[merged small][merged small][merged small][merged small][ocr errors]

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

[merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][subsumed][ocr errors][merged small][merged small][merged small]

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

[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][ocr errors][ocr errors][merged small][ocr errors]

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

[blocks in formation]

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

[merged small][merged small][merged small][merged small][ocr errors][merged small]

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

[ocr errors][ocr errors][merged small]

.. 0 : 0′ :: a + 2r: 250 × 7. (26r — a).

826.

upon

From the centre of the reflector let fall a 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 L, and whose focus is the centre of the reflector. Let 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,)

[merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]

and the equation between its rectangular co-ordinates is known, viz'

[ocr errors][merged small][merged small][merged small][merged small]

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

[merged small][merged small][merged small][merged small][ocr errors][ocr errors][ocr errors][merged small][merged small]
« ForrigeFortsett »