of the spectrum being between these extremes. An observer, therefore, whose eye is at E, will receive red light from the drop B and violet from the drop C and intermediate colors from drops between them. If the diagram is now conceived to be rotated about the line NES, joining the eye and the sun, it is clear that every drop on the circle whose radius is BN, which is described by the motion of the drop B, will send red light to the eye, while all drops on the circle described by the motion of C will send violet light to the eye. In this way a colored circular band will be seen whose angular radius is between 40° and 42°.

It will be observed also that the drop A does not send any light at all to the eye at E, while scattered rays of all colors come from the drop D. The region, therefore, above the primary rainbow appears dark, while that within it is bright, and the red of the bow is nearly pure, while the violet is mixed with scattered rays of other colors and fades out into white.

869. Secondary Rainbow.-For those rays that suffer twc reflections inside a rain drop there is also a certain direction in which the emergent rays are parallel, and therefore the light in that direction is particularly intense. In case of red light this bright pencil makes an angle of 231° with its original direction, while the bright pencil of violet light is bent around. through an angle of 234°.

A colored bow will, therefore, be produced as shown in figure 535, the angular radius of the red being 51° while that of the

Sun

violet is 54°. The sky within this bow near its red edge will be dark while above it beyond the violet the sky will be bright with scattered light.

870. Supernumerary Bows.-The bows caused by more than two internal reflections cannot be seen. A second and even a third band of red may, however, be occasionally seen in the violet region of the primary bow. These are called supernumerary bows and are diffraction phenomena ($928). Their explanation is given in more advanced treatises, such as Preston's Theory of Light.

OPTICAL INSTRUMENTS.

871. Optical Instruments.-There are two general classes of optical instruments, those which form a real image on a screen, as in case of the photographic camera and projection lantern (magic lantern), and those intended for direct eye observation in which the image formed is virtual. To the latter class belong the magnifying glass, microscope, and telescope.

To obtain a clear conception of the action of an optical apparatus it is desirable to study the effect of the instrument upon two pencils of light, starting from different points in the object and traced through to the corresponding points in the image. One pencil should be as oblique as can pass through the instrument.

872. Photographic Camera.—In the simplest form of photographic camera a single convergent lens forms a real image of a distant object on the sensitive plate. A diaphragm placed close to the lens limits the size of the pencil of light. The quickness of the lens or brightness of the image depends on the solid angle included in each pencil of light as it converges upon a point on the plate. If the area of the diaphragm opening is represented by A and the distance of the plate from the lens

A

f'

hence it is this ratio

by f the solid angle is proportional to which determines the necessary time of exposure, other things being equal.

Figure 536 represents a symmetrical or "rectilinear" lens consisting of two similar achromatic lenses, symmetrically

placed, and having the diaphragm half-way between them. It will be observed that in this case the oblique pencil passes as much below the center of the front lens as above the center of the back lens, so that the beam is as much bent by one as by the other and emerges parallel to the incident pencil. This

FIG. 536.

tends to cause straight lines in the object to be reproduced as straight lines in the image.

873. Distortion of Images.-The image of a grating with equal square openings may be distorted in either of the two modes shown in figure 537. The first or barrel-shaped distortion is seen when the center of the image is magnified relatively more than the outer portions, while the other form of distortion is caused by the greater relative magnification of the parts away from the center.

FIG. 537.

Either of these modes of distortion may be produced in projecting the image of the grating with the same lens, the form of distortion being determined by the mode of illumination. For let G (Fig. 538) be the grating and L the lens, then if the grating is illuminated by a beam of nearly parallel light, as direct sunlight, the light from the upper part

of the grating will pass through the upper edge of the lens L, and being bent down too strongly in consequence of the spherical aberration of the lens (§858) will come to focus at P farther from the center than P' and will thus cause the distortion shown in the second diagram of figure 537. If, on the other hand, by means of a convergent lens the

G

L

15

FIG. 538.-Parallel illumination. Pin-cushion-shaped distortion.

illuminating beam of light is converged so strongly that rays from the top of G are refracted by the bottom of the lens L, as shown in figure 539, the focus P will be too near the center and the distortion will be barrelshaped.

It is clear that the least spherical aberration and distortion will be

secured when the illuminating lens converges the light toward the center of the lens L, as shown in the diagram of the magic lantern, figure 540.

It is this same kind of spherical aberration which causes barrelshaped distortion in the photographic image when the diaphragm is placed (outside) in front of the lens; while if the diaphragm is behind the lens the opposite form of distortion results.

874. Projecting Lantern.-The optical system of the magic lantern, stereopticon, or projecting lantern is shown in figure 540. It consists simply of a front lens or objective L which forms a real image of the slide S on the screen at S'; and an illuminating system which consists of the source of light at E and the condensing lens C which converges the light through the slide S toward the center of the lens L.

Since the screen S' is usually at a considerable distance, the distance from the slide S to the lens L is nearly the focal length of the lens or lens combination. So that the width of the

image on the screen is to the width of the slide as the distance of

the screen is to the focal length of the front lens L. when the lantern is to be at a great distance from the screen a long-focus front lens should be used to prevent the image from being too large and dim.

Hence

The front lens is usually a combination of two lenses to secure flatness of field and freedom from color and distortion. The condensing lens consists of two plano-convex lenses with their convex surfaces almost touching. If the upper portions of the two condensing lenses are thought of as prisms, it will be noticed that with this construction each is nearly in the position of minimum deviation for the pencil of light passing through it; for the incident and emergent pencils make somewhat nearly equal angles with the two surfaces of each of the two lenses. Such an arrangement makes the spherical and chromatic aberration. very much less than if the lenses had been placed with their flat faces together, which would make practically a single doubleconvex lens.

875. Projecting Microscope.-For the projecting of micro