Consider two lenses A and B of focal length and placed in contact with each other.
An object is placed at O on the common principal axis. The lens A produces an image at and this image acts as the object for the second lens B. The final image is produced at as shown in figure.
PO = u, object distance for the first lens (A),
PI = v, final image distance and
, image distance for the first lens (A) and also object distance for second lens (B).
For the image produced by the first lens A,
For the final image I, produced by the second lens B,
Adding equations (1) and (2),
If the combination is replaced by a single lens of focal length F such that it forms the image of O at the same position I, then
From equations (3) and (4),
This F is the focal length of the equivalent lens for the combination.