An analytical study is made of the response of an astigmatic optical system to spatial frequencies in the object. Curves are given showing the response as a function of line frequency for varying values of astigmatism, the image plane being midway between the sagittal and tangential foci. A simple criterion for testing astigmatic systems is given. Tolerances on astigmatism and Petzval curvature are obtained. A comparison is made with results indicated by geometrical treatment, which shows that the geometrical treatment gives sufficiently accurate results only for low frequencies.