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Besides its energy range, the main characteristic of a particle collider is its luminosity. A very important piece of equipment to achieve high luminosities is a low-beta insertion, a device which focuses the beam down to a tiny spot size. The focusing limits are not only technological, they are also associated with the aberrations of such systems. This paper deals with the chromatic aberrations. In a first part, the treatment of the focusing errors is reviewed and a formalism using symbolic computation is set up so that high-order perturbations can be evaluated, although results are limited to the second-order betatron tune shift and to its interpretation. In a second part, a completely analytical theory of the final doublet of a low-beta insertion is given for a special model of flat-beam focusing.
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