44. If now we suppose Q to be very distant, a star, for instance, i. e. if we suppose a pencil of parallel rays to fall on the surface close to A and parallel to the diameter

through A, we may in our formula neglect

I

and we

where F is the position which the point q assumes in such As in reflexion, F is called the "Principal Focus."

a case.

Thus for parallel rays from air into water,

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AF3A0=4. AO.

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On the whole then we shall have, as in the corresponding case of reflexion, the following rules for determining the position of our Geometrical Focus.

(i)

and q must lie on the same normal, since all the rays from Q are supposed to proceed as if from q, after refraction, and since the direction of the normal ray is not altered at all.

i.e. The Geometrical Focus of any point must lie on the normal to the surface from that point.

according as the surface is concave or convex.

(iii) Q and q move in the same direction, since as AQ increases, so must Aq from the formula.

(iv) Hence and q always lie on opposite sides of the principal focus.

45. Let PQX be an object in front of the spherical refracting surface BAC.

fig. 38 (i), where

46. There is one more point to notice. In the formulæ for concave and convex refracting substances, we observe that the only difference is in the sign of the righthand side of the equation. The two formulæ will coincide, if we make this assumption about the signs of our lines. All lines measured from the origin A opposite to the direction in which the light is travelling are to be regarded as positive, and vice versa.

Applying this to figure 36, we see that all the lines there drawn from A are opposite to the direction of the incident light, and are all therefore positive. This case then may be regarded as the standard case.

But if the reader will draw the corresponding figure for a convex substance he will see, as in fig. 37 (ii), that AO is drawn from A in the negative direction. This accounts for the difference in sign in the two formula.

47. It is sometimes more convenient to have a formula referred to the centre as origin, instead of one referred to A.

From figure 36, Aq = AO+ Oq, AQ=AO + OQ.

surface. Fig. 36 is clearly the standard case for the formula origin O, since then all the lines OA, OQ, Og are positive.

48. We will, as a good example, trace the effects of refraction through a sphere of some substance denser than the surrounding medium.

Fig. 38.

B

P

We will suppose that the rays after each refraction come as if accurately from the Geometrical Focus.

(i) Let a small pencil of rays from the point P of the object PQ fall upon the surface BAC and be refracted into the sphere. After entering the sphere these rays will proceed as if they came from, on the normal op1 where

(ii) These rays, which come as if from p,, will, on being refracted out into the original medium, proceed as if they came from, where, since the refraction is from dense to rare,

To trace the course of the rays through the sphere to the

eye, we do just as in reflexion,

(i) Join the last image to the eye by straight lines and shade from the eye to the last surface.

(ii) Join the points where these lines cut the last surface, to the last image but one, and shade to the last surface but one.

(iii) Join these last points to the object and shade.

The shaded part represents the rays from the point P to the eye.

49. The three following examples are recommended to the reader's attention.

(i) The last example, supposing that instead of a sphere, the refracting solid is a hemisphere, and that the light falls first upon the convex surface.

(ii) A glass sphere is divided into two hemispheres, the index of refraction in one being different from that in the other. Find the positions of the images as in Art. 48.

(iii) An eye is placed close to the surface of a sphere of glass (μ = 3) which is silvered at the back. Shew

that the image which the eye sees of itself is natural size.

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