p. 354); the argument is based on the principle that the optical

distance from object to image is- constant. “ Taking the case of a convex lens of glass, let us suppose that parallel rays DA, EC, GB (fig. 14) fall upon the lens ACB, and are collected by it to a focus at F. The points D, E, G, equally distant from ACB, lie upon a front of the wave before it imP111§ @S UIJOH the lens. The focus is a point at which the different parts of the wave arrive at the same time, and that such a point can exist depends upon the fact that the propagation is slower in glass than in air. D A”

E

G B

FIG, 14.

amount of the retardation is measured by AF these retardation's must be equal, or AF-CF = The ray ECF is retarded

from having

to pass through the

thickness (ll) of

glass by the amount

(n- 1)d. The ray

DAF, which traverses

only the extreme

edge of the

lens, is retarded

merely on account

of the crookedness

of its path, and the

-CF. If F is a focus

(n-1)d. Now if y

be the semi-aperture AC of the lens, and f be the focal length CF, AF -CF = y/ (fl-l-yz) -f= sy'/f approximately whence Y,

f=i;v“/( -I)<i- (12)

In the case of plate-glass (n-1) =% (nearly), and then the rule (12) may be thus stated: the semi-aperture is a mean proportional between the focal length and the thickness. The form (12) is in general the more significant, as well as the more practically useful, but we may, of course, express the thickness in terms of the curvatures and semi aperture by means of d 2 %y2(r, -142-1). In the preceding statement it has been supposed for simplicity that the lens comes to a sharp edge. If this be not the case we must take as the thickness of the lens the difference of the thicknesses at the centre and at the circumference. In this form the statement is applicable to concave lenses, and we see that the focal length is positive when the lens is thickest at the centre, but negative when the lens is thickest at the edge.” Regulation of the Rays.

The geometrical theory of optical instruments can be conveniently divided into four parts: (1) The relations of the positions and sizes of objects and their images (see above); (2) the different aberrations from an ideal image (see ABERRA-TION); (3) the intensity of radiation in the object- and image spaces, in other words, the alteration of brightness caused by physical or geometrical influences; and (4) the regulation of the rays (Strahlenbegrenzung).

The regulation of rays will here be treated only in systems free from aberration. E. Abbe first gave a connected theory; and M von Rohr has done a great deal towards the elaboration. The Gauss cardinal points make it simple to construct the image of a given object. No account is taken of the size of the system, or whether the rays used for the construction really assist in the reproduction of the image or not. The diverging cones of rays coming from the object-points can only take a certain small part in the production of the image in consequence of the apertures of the lenses, or of diaphragms. It often happens that the rays used for the construction of the image do not pass through the system; the image being formed by quite different rays. If we take a luminous point of the object lying on the axis of the system then an eye introduced at the image-point sees in the instrument several concentric rings, which are either the fittings of the lenses or their images, or the real diaphragms or their images. The innermost 4

| . .

” = E4 ef

0' " 0'.

0

W,

Fig. 15.

and smallest ring is completely lighted, and forms the origin of the cone of rays entering the image-space. Abbe called it the exit pupil. Similarly there is a corresponding smallest ring in the object space which limits the entering cone of rays. This is called the entrance pupil. The real diaphragm acting as a limit at any part of the s stem is called the aperture-diaphragm. These diaphragms remain for all practical purposes the same for all points ying on the axis. It sometimes happens that one and the same diaphragm fulfils the functions of the entrance pupil and the aperture-diaphragm or the exit pupil and the aperture-diaphragm. Fig. 15 shows the general but simplified case of the different diaphragms which are of importance for the regulation of the rays. Sl, S2 are two centred systems. A"is a real diaphragm lying between them. B1 and B'2 are the httings of the systems. Then S1 produces the virtual image A of the diaphragm A' and the image B2 of the fitting B'2, whilst the system S2 makes the virtual image A” of the diaphragm A' and the virtual image B'1 of the fitting Bl. The object-point O is reproduced really through the whole system in the point O'. From the object-point O three diaphragms can be seen in the object-space, viz. the fitting B1, the image of the fitting B2 and the image A of the diaphragm A' formed by the system Sl. The cone of rays nearest to B2 is not received to its total extent by the fitting B1, and the cone which has entered through B1 is again diminished in its further course, when passing through the diaphragm A', so that the cone of rays really used for producing the image is limited by A, the diaphragm which seen from O appears to be the smallest. A is therefore the entrance pupil. The real diaphragm A' which limits the rays in the centre of the system is the aperture diaphragm. Similarly three diaphragms lying in the image-space are to be seen from the image-point O'-namely B', A", and B'2. A" limits the rays in the image-space, and is therefore the exit pupil. As A is conjugate to the diaphragm A' in the system Sl, and A" to the same diaphragm A' in the system SQ, the entrance pupil A is conjugate to the exit pupil A" throughout the instrument. This relation between entrance and exit pupils is general.

The apices of the cones of rays producing the image of points near the axis thus lie in the object-points, and their common base is the entrance pupil. The axis of such a cone, which connects the object point with the centre of the entrance pupil, is called the principal ray. Similarly, the principal rays in the image-space join the centre at the exit pupil with the image-points. The centres of the entrance and exit pupils are thus the intersections of the principal rays. For points lying farther from the axis, the entrance pupil no longer alone limits the rays, the other diaphragms taking part. In fig. I6 only one diaphragm L is

present besides the entrance

pupil A, and the object space

is divided to a certain

extent into four parts. The

section M contains all points

rendered by a system with

a complete aperture; N contains

all points rendered by

a system with a gradually

diminishing aperture; but

this diminution does not

attain the principal ray

passing through the centre

C. In' the section O are

those points rendered by a

system with an aperture

which gradually decreases to

zero. No rays pass from the

points of the section P

through the system and no 6

image can arise from them. FIG' I

The second diaphragm L therefore limits the three-dimensional object-space containing the points which can be rendered by the optical system. From C through this diaphragm L this three dimensional object»space can be seen as through a window. L is called by M von Rohr the entrance luke. If several diaphragms can be seen from C, then the entrance luke is the diaphragm which seen from C appears the smallest. In the sections N and O the entrance C";

A

luke also takes part in limiting the cones of rays. This restriction is known as the “ vignetting ”

action pf the entrar;ce lute? Tge

ase o the cone o rays or the »,

points of this section of the / 2 object-space is no longer a' circle // but a two-cornered curve which

arises from the object-point by %

the projection of the entrance / it

gtke on lthe elgitrance fpupili / / 17a s ows the ase o sue "fer"

a gone of rays. It often hap- gf/

pens that besides the entrance,

luke, another diaphragm acts

in a vignetting manner, then

the operating aperture of the cone of raYS 15 a 9“fV€ made UP of circular arcs formed out of the entrance DUPl1 and the two projections of the two acting diaphragrHS fflg- {7b)-If the entrance pupil is narrow, then the 5€CU0f1 NO, 111 Whlfih the vignetting is increasing, is diminished, and thefe S"f@allY °Y1lY QW? division of the section M which can be r€p1'0dLC6d, and Of the SCQQOH P which cannot be reproduced. The angle w+'w=2'w, Comprising the section which can be reproduced, is called the angle of the field of view on the object-side. The field of view 2°w retains its importance Fig. 17a. FIG. 17b.