In determining the position and magnitude of the image of an object we have hitherto only made use of certain functions of a lens or lens-system, and, moreover, considered the latter as perfectly free from all aberrations, including distortion.

Reference to the four "cardinal" points is of great value and simplicity in this respect, but it is obvious that these geometrical constructions do not in reality trace the actual course of the rays which form the images. This is particularly obvious in reference to the Telephotographic construction. (See Fig. 42.) The rays utilised to determine the position, etc., of the image may not even pass through the lens at all.

It may be the mounting of the lens itself or the aperture of the diaphragm which limits and determines the portions of the lenses themselves that are utilised in the production of the various points forming the entire image. In Fig. 45 (1) shows that rays at a certain obliquity to the axis cannot pass the lens at all, but are intercepted by the mounting of the lens; (2) shows that the diaphragm with a given aperture in one position allows the entire pencil of rays at a given obliquity to pass through both lenses, a definite portion of each combination being utilised to form the image; (3) shows that if the same diaphragm be placed in a different position the same pencil of rays meeting the lens at the same obliquity as in (2) is almost entirely cut off, but the figure shows that new portions of the positive and negative lenses are now utilised to form the image.

From diagrams (2) and (3) it is obvious that the size of the diaphragm will not only affect the different portions of the two lenses that are brought into play in forming the image, but also affect the illumination of the image, and the extreme obliquity at which a pencil of rays may pass through both lenses.

A glance at Fig. 46 will show the position in which the diaphragm should be placed in order to transmit the greatest angle; this is seen to be the crossing point of those rays which traverse both lenses from their extreme edges. The diaphragm now also occupies the position for the greatest equality of illumination. If we project the extreme incident rays to meet in the axis, they form an angle a, which determines the extreme angle included when no diaphragm is employed.

Let us now examine the effect of the diaphragm in greater detail.

On Rapidity. - In comparing the rapidity of two lenses, it is obvious that the one which gives an image of greater intensity, or possesses a higher degree of illumination than the other, is the more rapid of the two. In general we form an idea of rapidity of a lens by defining its Intensity ; this we will denote by I.

The intensity is the ratio of the diameter of the effective aperture to the focal length of the lens. Calling a the diameter of the effective aperture and f the focal length, the "effective" aperture. If a beam of parallel rays (Fig. 47) is incident upon a positive lens a, having a certain focal length f, the rays converge after passing through the lens and meet in a point f, its focal point. The diameter of a a is the effective aperture of the lens, as none of the rays incident upon it are intercepted after convergence. I f we place diaphragms bb, c c, of smaller diameter than a a, behind the lens as in (1), in positions which just allow the entire cone of rays a fa to pass, the diameters of b b and c c have the same effective apertures as a a. Calling a the diameter of a a, the intensity in each case is a/f.

I =a/f..........(20)

It is important that we should clearly understand the meaning of

Fig.46.

If, however, the diaphragm b' b' intercepts any of the rays transmitted through a a as in (2) the effective aperture becomes then a'a' ; calling b the diameter of a' a', then the intensity is b/f

For various reasons it may be necessary to reduce the diameter of the full effective aperture, and to accomplish this we make use of diaphragms whose effective apertures are arranged to have certain intensities. The intensities assigned to the diaphragms are indicative of the relative exposures necessary for each, and they are arranged so that the-latter are easily comparable. The term "stop" is synonymous with "diaphragm."

As the Telephotographic combination is used to give a great variety of focal lengths, it is most convenient to adopt a system of diaphragm notation that has direct reference to the focal length of its positive element.

Fig.47.

The stops are denoted by the ratio of the focal length of the lens to the effective aperture. Thus if f/a = 8, then the stop is called f/8 , or its a

• • / intensity is f/8.

V Fig. 48 shows the intensities (usually engraved on positive lenses) and relative rapidities of the diaphragm notation based upon the above system. In general, if the diameter of the aperture is known, with the focal length of the lens, and we express the intensity as above, the comparative exposures are as the squares of the denominators of the stop notation. Example: the relative exposures of stops f/10 and f/15 are as 102: 152, or as 1 : 2 1/4 (not given in the figure).

In the last chapter we have considered the Telephotographic lens from two different standpoints; the method of treating it (B) will readily commend itself for arriving at comparative exposures. It is only necessary to multiply the exposure required for the stop notation indicated on the positive lens by the square of the magnification, and we know the exposure to be given for the Telephotographic lens. The knowledge of the effective aperture in some Telephotographic constructions is most important because the stop is sometimes considerably removed from the front converging lens. The absolute measurement of its diameter would be very different from its real effective aperture, and if used in place of the latter as a basis of calculation would lead to very great discrepancies, and consequent failure in the correct timing of exposures.