This paper treats lens systems which consist of spherical and plane refracting surfaces, and for which fixed values of the indices of refraction have already been assigned. Simple algebraic equations are derived which allow changes to be made in the curvatures of two consecutive surfaces, together with changes in two specified vertex separations, subject to the following invariance condition: the heights and slope angles of both the marginal and principal paraxial rays must be left unchanged at all surfaces of the system other than the two surfaces undergoing modification. These equations may be considered to constitute a generalized “bending”, although no attempt is made to retain the Petzval contribution as an invariant. The use of a series of such “bendings”, applied independently at separate elements of the system, for certain adjustments of the third-order aberrations is immediately evident. Lagrange’s invariant allows a surprisingly concise formulation of the final equations. If the first surface of an element is located at the primary focal point of the second surface, a separate solution is derived in which only two lens parameters are to be changed. These results are apparently new, although similar results for single paraxial rays are well known. Tentative suggestions as to additional applications are made.
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