We may now proceed to consider the magnification to be given by the negative element with which it is combined to form the Telephotographic system. Let the negative element have a focal length of 6 inches, and suppose we wish to reproduce the object finally as half size, or in the scale 1/2. We must magnify the primary image 4 1/2 times ; to do this we set the screen 3 1/2 times the focal length of the negative lens from it, or at a distance of 21 inches. An adjustment between the separation of the two elements will enable us to focus accurately upon the screen in this position.

Any departure from the finest possible definition given by the positive lens (not exceeding 1/100 of an inch) will be magnified 3 1/2 times in the final image. This final degree of definition may or may not suffice, according to the requirements of the case. For a large portrait it will probably be sufficiently well defined, as the image will not be viewed too closely. On the other hand, if an absolute degree of definition is required through the depth of the image, equal to that in the primary image for example, it will be necessary to select an intensity which gives 4J times the depth necessary for the positive element. This is roughly given by dividing the intensity f/6 by the magnification to which the primary image has been subjected.

## Notes

On the "Pupils" of a Lens-system. - Let us first see how to construct the images of a diaphragm which are formed by the separate portions of a lens-system. In Fig. 55 the upper diagram consists of a lens-system composed of two positive lenses, L1, L2 respectively, the stop s being situated midway between them. In order to find the image of the stop formed by the lens l we draw a line parallel to the axis meeting the lens L1, which after refraction must pass through its focal point f1; again a ray from the edge of the stop s passing through the centre of the lens l1 continues in a straight line ; it will be seen that they meet in a point Sj; we thus find the image of any stop s formed by the lens l1 at s1 Similarly we determine the position of the image of the stop s formed by the component element of the system l3 situated behind the stop, and find its image in s2.

In the lower figure the front element of the system is a positive lens, and the back element is a negative lens ; we construct the image of the stop s formed by the lens l1 and find it at S1; and using the same construction as before for determining the image of the stop s by the negative lens l2, we find its image at s2.

It will be observed that in all cases the stop images in these constructions are virtual images. It is evident that s2 is the image of s1 formed by the whole lens-system, and s, and s2 are therefore conjugate to one another; so that any ray which passes through the image s2 must before refraction necessarily have passed through s1 and hence a stop or diaphragm in the position and of the size s, will have the same practical effect in limiting the pencils which form the image as a stop in the position and of the size of s2. Both have the same effect as the actual stop s.*

Fig.55

When we have determined the position of these stop images we are enabled to trace the actual course of the rays which pass through any lens-system by an accurate and pretty geometrical construction, and thus determine the image of any object. Referring to Fig. 55A, if

* Dr. S. Czapski, "Theorie der optischen instrumente nach Abbe," pp. 155, 156.

## Table Of Front And For An Object At

 DISTANCE OF OBJECT. 3 FEET. 4 FEET. 5 FEET. 6 FEET. 8 FEET. Focal length of Lens in inches. Intensity I. Scale, or1/n-. Front depth in inches. Back depth in inches. Scale, or 1/n. Front depth in inches. Back depth in inches. Scale, or 1/n. Front depth in inches Back depth in inches Sca'e, or 1/n Front depth in inches. Back depth in inches. Scale, cr 1/n . Front depth in inches. Back depth in inches. *6" F/3 I/5 .9 .93 1/7 l.6l 1-75 I 8 2.05 2.26 1/11 3-73 4.21 1/15 6 1/2 7 3/4 F/4 1/15 1.25 2.T3 2.36 2.71 3.06 4.8 54 84 I0 3/4 F/5 1.43 1.6 2.62 3 3-35 3.89 6 7i 10 1/2 13 3/4 F/6 1.7 i-9 3.1 3-65 3-96 4.76 7 9 12 1/2 17 1/4 F/8 2.22 2.6 4.05 5-01 Si 6 1/2 9 "4 15 3/4 24 1/2 F/10 2-7 3.33 4.94 6 1/2 6 1/4 8 1/2 11 16 1/2 19 32 3/4 *8 1/4 F/3 1/3-3 • 42 .43 1/4.8 .81 •85 1/6.3 1-34 1.42 I 8 2.09 2.23 I II 3.79 4.14 F/4 4 •55 .58 1.08 1.14 1.77 1.9 2 75 3.01 4.99 54 F/5 .69 .73 1-34 1.44 2.2 2.4 3.41 3-8 6 1/4 7 F/6 .82 .87 1.6 1.74 2.62 2.91 4.05 4.62 7i 8 3/4 F/8 1.09 1.24 2.1 2.4 3.43 3.96 5 1/4 6 1/4 94 12 F/10 1.35 1.49 2.6 2-99 4.22 5.04 6 1/2 8 11 1/2 15 1/2 *10" F/3 1/2.6 .28 .29 1/3.8 •54 •55 1/5 .88 .91 1/6.2 1-31 1.37 I/8.6 2.41 2.55 F/4 •36 .38 •71 •74 1.17 1.22 1.73 1.83 3.18 3.43 F/5 •45 •47 .89 •93 1.45 1.54 2.15 2.31 3-94 4.33 F/6 •55 •57 1.06 1.12 1.73 1.86 2.56 2.8 4.68 5.26 F/8 .72 •77 1.4 1.51 2.28 2.52 3.37 3.79 6 1/4 7 1/4 F/10 •9 •97 1.74 1.91 2.83 3.19 4.16 4.81 7 1/2 9 1/4 *I2" F/3 1/2 •17 .18 1/3 •35 •36 1/4 •59 .6 1/5 .88 •9i 1/7 1.64 1.71 F/4 •23 .24 •47 .48 •78 .81 1.17 1.22 2.18 2.3 F/5 .29 •3 •59 .6 .98 1.02 1.46 1.54 2.71 2.85 F/6 •35 •36 •7 •73 1.17 1.23 1.74 1.85 3.23 3.5 F/8 •47 •49 •93 .98 1.55 1.6S 2.31 2.5 4.26 4.73 F/10 .58 .61 1.16 1.24 1.92 2.08 2.85 3.15 5 1/4 6

* These particular focal lengths are chosen as they correspond to those of well known into Telephotographic instruments

## Back Depth Of Focus. Given Distances

 10 FEET. 12 FEET. 14 FEET. 16 FEET. 18 FEET. 20 FEET. 24 FEET. Scale, or - 1/n. Front depth in inches. Back depth in inches. Scale, or -.1/n. Front depth in inches. Back depth in inches. Scale, or - 1/n. Front depth in inches. Back depth in inches. Scale, or -. Front depth in inches. Back depth in inches. Scale, or -. Front depth in inches. Back depth in inches. Scale, or 1/n. Front depth in inches. Back depth in inches. Scale, or -. » Front depth in inches. Back depth in inches. 1/19 101/2 123/4 1/23 143/4 18 3/4 1/27 20 261/4 1/31 253/4 351/2 1/35 32 46 1/39 39 58 1/2 1/47 544 89 131/2 I71/2 19 261/4 254 371 /4 323/4 501/2 403/4 66 1/4 491/4 85 68 1/2 132 3/4 161/4 223/4 23 344 30 3/4 49 1/4 391/4 674 481/2 90 58 1/2 117 80 1/2 188 19 28 1/2 26I 434 354 63 45 1/4 874 554 18 67 156 91 1/2 260 1/4 24 41 1/2 334 65 44 96 1/2 554 138 1/2 68 194 81 1/2 267 1/2 110 501 1/3 28 1/2 57 39 1/2 92 511/2 141 3/4 64I 2121/2 78 3/4 315 934 468 125 1/3 1128 1/14 6 6 3/4 I/17 8 1/2 9l I 20 11 3/4 135/8 1/23 15 1/4 18 1/26 19 1/4 234 1/29 23 12/ 29 1/4 1/35 334 434 7 3/4 9 11 1/4 13 1/2 15 1/4 18 3/4 19 3/4 25 24I 32 1/3 30 1/4 40 3/4 43 61 9S 11 1/2 13 3/4 17 1/2 18 5/8 24 24 321/4 3°i 42 36 3/4 53i 51 3/4 80 1/2 11 1/4 14 1/8 16 1/4 21 2I| 30 281/4 401/8 35i 52 1/2 42 3/4 66 3/4 60 102 1/2 14 1/2 19 1/2 20| 29S 28 42 1/4 357/8 574 444 76 54 98 1/4 74l 154l3/4 17 3/4 25 3/4 25 39 334 56 1/2 423/4 78 53 104 1/4 63 3/4 136 3/4 873/4 2231/2 1/11 3.82 4.11 I/13.4 5 1/2 6 1/15 7 74 1/18 93/4 11 1/20.5 12 1/2 14 1/4 1/23 151 1/2 17 3/4 1/29 24 28 3/4 5.04 5.54 7i 8 1/4 9 10 1/4 123/4 14 3/4 16 1/4 19 1/4 20 24 1/2 31 394 6 1/4 7 9 10 1/4 11 13 151/2 19 20 243/4 243/4 31 1/4 38 51 1/4 74 8 1/2 10 2/2 I2| 13 16 181/2 231/4 231/2 301/4 29 38 3/4 441/4 63 1/2 94 11 3/4 13 3/4 17 1/2 17 22 233/4 321/4 30 42 1/2 37 544 56 91 1/2 11 3/4 15 16 3/4 22 1/2 20 3/4 28 1/2 28| 424 36 1/4 56 444 72 1/2 67 124 1/2 1/9 2.63 2.77 1/II 3.84 4.08 1/13 5 1/4 54 1/15 7 74 1/17 8| 94 1/19 11 12 1/23 151/2 17 1/2 3.48 3.72 5 5 1/2 6 3/4 74 9 10 11 12/ 13 141/4 161/4 201/2 24 4.32 4.79 61/4 7 84 94 111/4 12 3/4 14 1/4 161/2 171/2 20 3/4 25 30 1/2 5.14 5.69 71/2 8 1/2 10 1/4 11 1/2 131/4 15 3/4 16| 3/4 20 20 3/4 25i 29 1/2 374 6 3/4 7 3/4 93/4 111/2 13 153/4 173/4 211/2 213/4 273/4 26 3/4 35 38 52 1/2 81/4 93/4 12 141/2 16 201/4 211/4 273/4 261/2 36 32 1/2 454 46 69

Carte de Visite and Cabinet lenses of high intensity, and are well suited for conversion for taking large portraits in the studio.

from a point o in the object plane o1 oo2 we draw a pencil of rays to the edge of the image of the stop s formed by the front lens at s1 it will be found to cut the lens l: at the points a a ; the actual course of the rays, after passing this lens, will be through the edge of the stop s itself, in the direction a b, a b meeting the lens l2 at b b. These rays emerge from the lens l, as though they had proceeded from the image of the stop s formed by the back lens at s2, and meet in the point i of the image. Professor Abbe defines the image of the stop s formed at S1 which appears from o under the smallest angle (2 u) as the "entrance pupil" (eintritts pupille) of the system ; similarly the stop whose image s2 appears from 1 under the smallest angle (2 u) he terms the (austritts pupille) "exit pupil."

In a similar manner to that employed for tracing the course of the pencil of rays from o through the entire system, we may construct the image formed by any points o1 o2 in the plane of the object forming images at I1 I2 respectively; the dotted lines in the figure indicate the construction. It is evident that o1 and o2 may occupy positions such that rays proceeding from objects further from the axis cannot emerge from the lens at all ; when o1 and o2 are situated in such positions that rays proceeding from them can only just emerge through the entire system, o1 o2 will appear from the centre of the "entrance pupil" s1 under an angle 2 w ; 2 w determines the angle of the field included by the lens. The image I1 I2 forms an angle 2 w1 from the centre of the " exit pupil" s2, and appears from it under the smallest angle.

We thus find that all the pencils of a system are of two kinds :

(1) those which have their common base in one of the "pupils" and their apices in either the field of the object or the field of the image ;

(2) those which have their common bases in either the object or the image, and their apices in the corresponding "pupil."