Combination of Thin Lenses in Contact

Many real optical instruments (a camera lens, a microscope objective, a telescope eyepiece) use not one single lens but two or more thin lenses placed close together, in contact, on a common axis, in order to combine their individual converging/diverging effects and, in well-designed combinations, to correct for the aberrations any single lens would otherwise introduce. For two or more thin lenses of focal lengths $f_1,f_2,f_3,\ldots$ held in contact, the equivalent focal length $F$ of the whole combination, acting as a single thin lens, is found by applying the thin lens formula in turn at each lens (treating the image formed by one lens as the object for the very next one) and adding the resulting equations; the intermediate image distances cancel exactly as they did within the derivation of a single thin lens's own formula, leaving simply $\frac{1}{F}=\frac{1}{f_1}+\frac{1}{f_2}+\frac{1}{f_3}+\cdots.$ Equivalently, and often more directly useful, the powers of the individual lenses simply add: $P=P_1+P_2+P_3+\cdots.$ This additivity of power is precisely why power, rather than focal length, is the natural quantity for an optician to prescribe and for a lens designer to specify: whether the overall combination converg …