This section is from the book "Telephotography: An Elementary Treatise On The Construction And Application Of The Telephotographic Lens", by Thomas Rudolphus Dallmeyer. Also available from Amazon: Telephotography and Telephotographic Lens.
Let us now consider the Telephotographic lens as consisting of two separate parts.
The positive element may be considered as forming an image of definite size, of a given object, dependent upon its focal length, and the negative element may be considered as enlarging the image which the positive lens would have formed by itself.
In general, if one positive lens is n times the focal length of another we have seen that the image produced by the one will be n times as large as that produced by the other. From this it is evident that if we increase the size of any given image n times, this enlarged image will be identical with that produced by a lens whose focal length is n times the focal length of the lens that produced the original image.
In Chapter III (The Formation Of Images By Positive Lenses). we have ascertained how images of either near or distant objects are formed by a positive lens; and in Chapter IV (The Formation Of Images By Negative Lenses). we found the law for finding the "magnification" of the image of an object by a negative lens: "Divide the distance of the object from the lens, by the focal length of the lens and add one." The image formed by a negative lens alone is virtual, and as the above law shows us that a diminished virtual image is formed by a negative lens of a real object we must invert the conditions and form a real image of a virtual object! (Image and object are always interchangeable )
Photograph (by 10-in. stigmatic) of Boscombe Gardens in foreground and the Isle of Wight in the distance. The " Needles" in the centre are not visible, as the scale they are rendered in is too small; the cliff can just be distinguished. Taken 3 p m., April 19, 1899.
(By the Author.)
Photograph from the same standpoint as Plate XI., and at the same time by telephoto lens, composed of 8 1/4. de v. as positive element, and 2-in. negative. Camera extension from negative element 18 in. Exposure seven seconds ; yellow screen and isochromatic plate.
(By the Author.)
This is precisely what we do accomplish in the Telephotographic construction.
The positive lens l1 of focal length f1 would form a real image at a b, but the rays are intercepted by the negative lens l2 of focal length f1 We may then consider ab a new "virtual object," a real image of which a' b' is formed by the negative lens l2.
In this method of treating the subject we shall always refer the size of the final image a' b' to the size of the image a b which will be formed by the positive lens alone.
We repeat that in speaking of relative sizes, we refer to linear "magnification." unless otherwise stated.
In the Telephotographic lens we may place the screen at any distance we choose from the negative lens, and in order to find how many times we have magnified the image formed by the positive lens alone, we have seen that we must divide this distance by the focal length of the negative lens and add one.
Calling m the magnification, and e the camera extension (distance between negative lens and screen), and f2 the focal length of the negative lens :
M = E/f + I........(12)
Rule. - To find the magnification : divide the camera extension by the focal length of the negative lens and add one.
Rule. - To find the camera extension necessary for a certain magnification: multiply the focal length of the negative lens by the magnification less one.
E =f2(M - I).......(13)
These two rules apply for either near or distant objects. For a near object the focal length of the positive lens may be said to be temporarily increased and the image of a near object produced by it is larger than when the object is distant, obeying the law of conjugate foci. The conditions for the magnification of this image are not, however, interfered with.
If the object is very distant (the position of the focus of the positive lens being that of its focal point) we can at once derive the focal length of the Telephotographic combination for any given magnification. It is simply
= mfi; f = mf1.......(14)
Rule. - To find the focal length of Telephotographic lens for any chosen extension of camera: multiply the focal length of the positive lens by the magnification.
This may also be found in a perhaps simpler manner from the following consideration. As the positive lens f1 is some multiple of the negative lens f2, if we call this m, then m =f1/f2, and the focal length of the combination for a distant object,
F = mE +f1........(15)
Rule. - The focal length of a Telephotographic lens is equal to m times the camera extension plus the focal length of the positive lens. Let us now apply these rules to the example illustrated in Fig. 43.
Here f1 = 6 inches; f2 = 3 inches ; e = 9 inches, and f1/f2 = m = 2.
Hence, m = 9/3 + 1 = 4 ; and
E = 3 (4 - 1) = 9, or the image is four times as large as that given by the positive lens alone, and it is evident that: f = 4 x 6 = 24 (from (14) ); or again, f = 2 x 9 + 6 = 24 (from (15) ). 71
We thus see that with a camera extension of only 9 inches from the negative lens, we can obtain an image of the same size as that given by an ordinary positive lens of 24 inches focal length, requiring this length of camera !
In other words, the true focal length is the distance between p and f, although the distance between v" and f is all that is necessary to obtain it. As in ordinary positive systems, p is one of the principal points and f one of the focal points of the Telephotographic system.