Similarly it will be seen that in the convex mirror the image is smaller and erect.

21.

The Geometrical Focus of a pencil of rays from an infinite distance—¿.e. a pencil of parallel rays, is called the Principal Focus" of the mirror.

I

Now when AQ is infinitely large, is indefinitely

AQ

small and may be left out, and therefore the formula

I

==

I

2

+

becomes

AQ Aq AO

AF AO

where F is the prin

cipal focus. That is, F is midway between A and O, or the image of a star would be seen there, since a star is a very long way off.

22. There is now one point to notice to which we shall often again refer, regarding the signs of lines. We will always make use of the following convention. All lines measured in the direction opposite to that of the incident light are to be regarded as positive lines, and vice versa. Thus in Fig. 16 AO is positive, and therefore that is taken as the standard case. AQ must, of course, always be positive for divergent pencils. Now consider the case of a convex mirror with a divergent pencil. AO is then measured in the same direction as the incident light and is therefore negative. The formula, therefore, which for the concave

mirror was

(Compare Art. 47.)

This point will present no difficulty to those readers who are familiar with the elements of Analytical Geometry. Suppose now that we wish to find the position of the Principal Focus. This is the Geometrical Focus of a pencil

of rays from an infinite distance. Thus if Fis the Principal

I

AF AQA O'

2

AAO.e. the Principal Focus lies halfway between the mirror and its centre-or the light from a star would make a bright spot there.

23. Thus, finally, we have the following facts connecting Q and 9.

(1)

and lie on the same normal. For all rays from when reflected pass through 7, and since the normal ray is reflected along itself, 9 must lie

upon it.

(2) If A be the extremity of the axis, and the pencil

(3) Since

2

AO

= constant quantity, as AQ increases or decreases, Aq does just the opposite, in concave mirrors; i.e. Q and q move in opposite directions. In the convex mirror they move in the same

direction.

(4) Since, in the concave mirror, when

goes farther from A, q comes nearer, and, since, when Qis at its farthest, q comes to F the principal focus, therefore Q and a must lie on the same side of F always. Similar reasoning proves the same fact for the con

vex mirror.

By a principle which we have used before (Art. 15), if light can travel from any luminous point Q to any other

point q, by any number of reflexions, then if a light be placed at q its rays will go to Q by precisely the same path that it used before. This would be the case in fig. 16. Or

I

we can see it at once from the equation +

I

=

2

Aq AQ AO' where it makes no difference if we interchange AQ and Aq.

24. We will investigate the motion of Q and q still further.

if AQ decrease, Aq must increase (fig. ii).

(3) Let Q come to O the centre. The reflected rays all converge to O. Q and g meet (for if AQ=AO so does Aq).

(5) As still approaches A, q as it were turns the corner at infinity and appears again to the left of A, approaching A (fig. iv).

(iv)

५

(6) When arrives at A, q does so also; for from

CHAPTER IV.

CAUSTIC CURVES, &c.

25. IT cannot be too often recollected that the point q is not the point through which all the reflected rays pass, for there is no such point. But close to it the rays which fall on the mirror very close indeed to the axis, from Q, cut the axis after reflexion. If we take rays from Q which fall on the mirror further from the axis, we find that they cut the axis at appreciable distances from q; and also that the points of section of the axis become more and more scattered as the rays fall on the mirror further and further from the axis, all being on the side of q towards A.

Thus, if the dotted lines represent the reflected rays,