and it is now clear that the angle b Fd, that is to say, the divergence of the reflected ray from its original direction is proportional to the distance k of the point of incidence from the axis of the mirror.

The angle bFd may, however, be expressed in another way; for example, it may be said

or again, because on account of the smallness of the angle bFd the line b F is scarcely different from the focal distance d F, which we indicate by f,

This expression, compared with the above, leads to the equation

which enables the position of the point F, where the reflected ray cuts the axis, to be determined. But since the magnitude k, because it appears as a factor on both sides of the equation, may be eliminated, it is obvious that the position of the point of incidence ẞ has no influence upon the determination of the point F;

Determination of the position of conjugate points.

that is to say, all rays coursing parallel to the axis pass after reflexion through one and the same point F, situated upon the axis, the distance ƒ of which from the mirror is determined by the equation,

The focal distance is consequently equal to half the radius.

If we now consider any ray A b, proceeding from the point A, making (fig. 37) the angle a with the axis, we shall find that it is so reflected in the point b that the angle of incidence and the angle of reflexion are both = d, and the reflected ray cuts the axis at the point B at an angle ß. If now the angle which the axis of incidence drawn towards b makes with the axis be indicated by y, we obtain, because ẞ is the external angle of the triangle BCb and y is the external angle of the triangle CAb, the two equations,

B = x + d

a = y — d,

which added together make

a + b = 2y;

that is to say, for every point of the mirror the sum of the angles which the incident and the reflected ray make with the axis is inalterable, and is indeed equal to the deflection which the ray passing to the focal point experiences at the point.

If now the focal length of the mirror be indicated by f, and its radius consequently by 2f, and further, the distance of the luminous point d A (=b A) by a, the distance dB (= b B) of the image-point by b, and the perpendicular let fall from the point of incidence b upon the axis, by k, we obtain from the above-mentioned method of measuring the angles,

and consequently if these are arranged in the equation a+6 = 2y

This very circumstance, that the magnitude k, which alone refers to the position, whatever that may be, of the point of

incidence, is removable from the equation, supplies the proof that all rays proceeding from the point A, wherever they may strike the mirror, are united in the selfsame point B.

From the form of this equation, which expresses in the simplest manner the opposite relation of two conjugated points, it is further evident that the light-point and the image-point are mutually interchangeable.

The deviation which the ray incident in b experiences is 20. But from the above equation, it results that

The accuracy of the statement above made, that the deflections which the rays proceeding from any point experience are proportional to the distances of the points of incidence from the axis of the mirror, is thus rendered evident.

In order to determine the position and size of the image by construction it is not necessary to draw a great number of rays, as in figs. 31, 32, and 34; but only two rays for each point of

the image, because the others necessarily meet at point where these decussate. The two rays selected should be such as to make the construction as neat and convenient as possible. In fig. 38 the object whose image is to be determined is a straight line A a, perpendicular to the principal axis. Let the secondary axis AC be drawn to the point A; the ray coursing in this axis is of course reflected upon itself. Now let the ray parallel to the principal axis be drawn; this passes after reflexion through the principal focus, and the image of the point A required lies at the point B, where it cuts the secondary axis A C, and if Bb be

let fall perpendicularly to the chief axis we obtain in Bb the image of the object A a.

The course of all other rays proceeding from A may now be followed with facility. Thus, for example, the ray A o, which strikes the centre of the mirror o, is reflected in the direction o B. And as at the point o the principal axis is the axis of incidence, the angle Ao a is equal to the angle Bob. If the magnitude of the object A a be indicated by the sign p, the magnitude of the image Bb by the sign q, and the distances of the object and of the image from the mirror as before by the signs a and b, it is clear that

p: q=a: b;

that is to say, the size of the object stands in the same relation to the size of the image as the distance of the former from the mirror is to the distance of the latter from the mirror, a proposition that holds equally for the virtual as for the real image. The equations that have been deduced in the case of concave mirrors hold also for convex ones, if the virtual focal distance be regarded as negative, that is to say, as finstead of f.

CHAPTER V.

REFRACTION.

29. THE adjoining figure (fig. 39) represents a cubic vessel the sides of which are made of glass. A beam of parallel rays of light from the sun directed horizontally into the room by means of the Heliostat

FIG. 39.

Refractor.

is thrown obliquely upon the surface of the water by a small mirror. A part of the rays is, in accordance with known laws, reflected at the surface of the water, whilst another portion penetrates it; this last, however, does not pursue a course directly continuous with the incident rays, but follows a steeper, though still always straight direction.*

* The course of the incident and reflected rays of light in the air is readily recognised by the illumination of floating particles of dust, and in