Irreducible aberrations of a lens used for a range of magnifications

In practical photography a lens must form a high-quality image at a variety of magnifications. It has, however, been known for more than a century that no lens can be designed that is perfect at more than one conjugate. It follows that lenses designed for a range of magnifications must have residual aberrations that cannot be corrected no matter how complicated the lens is made. In this paper it is shown how these residual aberrations can be calculated.