The interval O'O", termed the astigmatic difference, increases, in general, with the angle W made by the principal ray OP with the axis of the system, i.e.
It is seen (ignoring exceptional cases) that the pencil does not meet the refracting or reflecting surface at right angles; therefore it is astigmatic (Gr.
If the above errors be eliminated, the two astigmatic surfaces united, and a sharp image obtained with a wide aperture - there remains the necessity to correct the curvature of the image surface, especially when the image is to be received upon a plane surface, e.g.
Spherical aberration and changes of the sine ratios are often represented graphically as functions of the aperture, in the same way as the deviations of two astigmatic image surfaces of the image plane of the axis point are represented as functions of the angles of the field of view.
To the greater power belongs the weaker dispersive power (greater v), that is to say, crown glass; consequently the crown glass must have the greater refractive index for astigmatic and plane images.
Two " astigmatic image surfaces " correspond to one object plane; and these are in contact at the axis point; on the one lie the focal lines of the first kind, on the other those of the second.
Systems in which the two astigmatic surfaces coincide are termed anastigmatic or stigmatic.
Sir Isaac Newton was probably the discoverer of astigmation; the position of the astigmatic image lines was determined by Thomas Young (A Course of Lectures on Natural Philosophy, 1807); and the theory has been recently developed by A.