If f represents the focal length of an ordinary positive lens, nf any multiple of it,

By the law of conjugate foci: f2 =xy..........(1) where x=nf, and y=1/nf.

If o be the distance of any object from the lens, and i the distance of the image from it: o = (n +1)f i=(n+1/n)f

.......(2)

## Telephotographic Lens (A)

If f represents the focal length of the combination, f1 and f2 the focal lengths of the positive and negative lenses composing it, d an interval of separation greater than the difference of their focal lengths, and a their entire separation : f=f1f2/d..........(3) and the back focal length bf =f2(f1-a)/d........(4)

Calling m the ratio of the focal lengths of positive and negative elements composing the system, or its power: or m = f1/f2 f1=mf2

..........(5)

.........(6)

If o be the distance of any object from the Telephotographic lens, and 1 the distance of the image from it :

0 = nF + mF +f1.......(7)

I =I/n F + I/m F - f2............(8) where n, as above, is any multiple of the focal length of the entire system, and m the focal length of the positive element divided by the focal length of the negative element.

To compare the distance between object and lens, and image and lens in the case of an ordinary positive lens, and in the case of the Telephotographic lens; by equations (2) (7) and (8), we see :

0 = 0 + {F(m - 1) +f1}......(9) .

I = i- {F(I - I/m) -f2}......(10)

Calling d the interval of separation between the positive and negative elements greater than the difference of their focal lengths : Separation of principal points

= 1/d(f1-f2+d)2......(11)

## Telephotographic Lens (B)

Calling m the magnification to which the image formed by the positive element f1 is subjected by the negative element f2 and e the camera extension from the negative element to the screen :

M = E/f2 + I.........(12)

E =f2(M - l)........(13)

If f represents the focal length of the entire system, and m, f1/f2, as before,

F = Mf1.........(14)

F = mE+f1.......(15)

If/ is the correct exposure to give to the positive element f, if used alone, and t that for the entire system: t = m2 t.........(16)

If - represent the scale of the image given by the entire system, and - that given by the positive lens alone

1/N = 1/n M..........(17)

To find the focal length of the entire system when a near object is rendered in a given scale of -,

F = mE+f1/m.........(l8)

It is evident that when N is 00, this equation is identical with (15). If we wish to reproduce an object in a given scale - by the entire system of definite focal length F,f1 and f2 being known, and we propose to magnify the primary image M times (for convenience), in order to do so, we see from (17) that n= mn, and hence, distance of object from lens

= (mn + 1)f1........(19)

## The Diaphragm

If a represents the effective aperture of the lens,/ its focal length, and 1 its intensity,

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

Calling e the camera extension, f1 and f2 the focal lengths of positive and negative lenses, and d1 and d2 their diameters respectively, and d the circle of illumination covered by the system :

D = E/f2(a1f2 + a2f1/f1-f2)........(21)

Using the same notation as before, and taking 1/100 of an inch as limit of indistinctness :

= 100 af + f........(22) or

= 100 If2 +f........(22a)

For a near object accurately focused :

Front depth,

= nf(n +1)

100 1/+ (n + 1) ..(23)

Back depth,

= »/(n +I) ....... (24) iooif-(n + 1)