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7. A small plane area is placed parallel to a plane lamina of intrinsic brightness I, of breadth 2a, and of infinite length, at a distance c from the centre of the lamina in a line dicular to the lamina. Prove that the illumination at the παι centre of the plane area is √a2 + c2

8. Show how to calculate the illumination produced by a window on a point of the floor directly in front of the centre of the window: the window being supposed to reach to the level of the floor.

9. Two spheres are luminous, and a small plane area is placed on a line joining their centres, its plane being perpendicular to this line. Find where it must be placed in order that its two surfaces may be equally bright.

10. Three equally bright points are placed at the angular points of an equilateral triangle. If a plane area be placed at the centre of the triangle in any manner, show that it will be equally bright on both sides.

11. A triangular prism, whose nine edges are all equal, is placed with one of its rectangular faces on a horizontal table, and illuminated by a sky of uniform brightness; show that the total illuminations of the inclined and vertical faces are in the ratio of 2√3 to 1.

12. A luminous point is placed on the axis of a truncated conical shell; prove that the whole illumination of the shell varies as

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where a,, a, are the radii of the circular ends of the shell, and c, c2 the distances of the luminous point from their planes.

13. The sides of a triangle are the bases of three infinite rectangles of the same brightness, whose planes are perpendicular to the plane of the triangle: show that all points within the triangle are equally illuminated. Find the position of a point in the plane of the triangle, such that the illuminations at that point received from the three rectangles may be equal.

14. In making with an astronomical telescope an observation for which it is essential that the brightness of the image on the retina should be at least a hundredth part of that of the object, show that the highest magnifying power that can be obtained is 1000, the diameter of the object-glass being 25 inches and that of the pupil of the eye inch. What is the highest magnifying power that can be used without any diminution of brightness?

CHAPTER XI.

THE RAINBOW.

136. In this Chapter we propose to give a brief explanation of the formation of a Rainbow, as far as it can be done by the principles of Geometrical Optics.

137. If a pencil of parallel rays falls on a refracting sphere, some portion of the light will be reflected externally, some portion will be refracted into the sphere. Of this latter part, when it is incident internally at the surface of the sphere, some portion will emerge, and another portion will be reflected internally, and be again incident on the internal surface of the sphere. At this second incidence the same division will again take place, and so on, at each successive internal incidence.

The primary and secondary rainbows are produced by portions of sunlight which, having been incident on raindrops, emerge after one or two internal reflections respectively.

We have therefore to consider mainly the circumstances attending the refraction and reflection of these portions in the case of light incident on a sphere of water.

138. Let PQ be the axis of a pencil of parallel rays incident on a refracting sphere at Q, refracted at Q along QR, reflected internally at R, and again incident at S. Let ST be the direction of that part which emerges at S, and

A. G. O.

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SU the direction of the reflected part, which is incident internally again at U, where some part of it emerges along UV.

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We shall first examine the positions of the primary foci of the pencils emerging at S and U respectively.

Let be the angle of incidence at Q,' the angle of refraction. It is plain that will also be the angle of incidence at R, S and U, and that will be the angle of emergence at S or U.

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Let be the index of refraction, r the radius of the sphere. Let q1, 92, 9, be the primary foci after refraction at Q, reflection at R, and emergence at S respectively. Let 94, 9s be the primary foci after reflection at S and emergence at U respectively. Then, by Art. 46, since the incident pencil consists of parallel rays,

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cos2 o'

μ cos p' — cos'

But QR=2r cos d'; . Rq, 2r cos p'

=

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μr cos2 d'

μ cos - cos

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By proceeding in this way it will be easy to obtain a second result

Uqs

r cos & (μ cos έ' – 6 cos p)

2 (u cos p'-3 cos 4)

(2).

We may notice incidentally that Sq, and Uq, respectively become infinite, that is, the rays in the primary plane emerge as a pencil of parallel rays after one or two internal reflections, when

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139. We shall now restrict ourselves to the consideration of the portion of light which emerges after one internal reflection.

The deviation of the axis of the pencil at Q is clearly -': at R its deviation is π-26'; and at S it undergoes a farther deviation in the same direction of deviation on the whole is therefore

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-'. Its

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