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XIX. On Astigmatic Lenses. 

 By K. J. Sowter, £.&>., A.R.C.Sc* 



A LENS which so acts on light falling on it as to produce 

 two focal lines in the refracted-ray system is termed 

 an astigmatic lens. Sturm (Comptes rendus de V Academie 

 des Sciences de Pa?is, t. xx.) investigated the refraction of a 

 circular pencil of light by an asymmetrical surface or lens 

 with circular aperture, and showed that for a convex refract- 

 ing lens the ray surface was a skew surface {surface gauche) 

 bounded by two right lines. Fick (Mediz. Pliysik) has 

 written on the subject; and Knapp [ArcJiiv f. Ophthalmologic, 

 Band viii.) has mathematically determined the form of the 

 whole refracted-ray surface. Donders, Reusch, and others 

 have investigated the optical properties of asymmetrical or 

 astigmatic lenses; and more recently Prof. S. P. Thompson 

 has, in his paper on " Obliquely-crossed Cylindrical Lensesf," 

 deduced a very simple geometrical construction for the 

 sphero-cylindrical lens equivalent to two obliquely-crossed 

 cylindrical lenses. 



The intent of this paper is to simplify the systematic 

 investigation of the properties of astigmatic lenses, and to 

 show the relation between a general type of astigmatic lens 

 and its equivalents. 



The quadric surface is the surface of lowest degree which 

 is capable of representing the general type of thin astigmatic 

 lens. An ellipsoidal lens is here considered as the funda- 

 mental form or type. Thin lenses only are treated. 



I. Ellipsoidal Lenses. 



A system of parallel and axial rays of light after passing 

 through an ellipsoidal lens becomes a system of rays inter- 

 secting in two straight lines. These lines are at right angles, 

 are the focal lines of the lens, and are separated by an interval 

 —the focal interval of Sturm — which depends on the ellipticity 

 of the lens. The focal lines are parallel to the elliptic axes of 

 the lens, and correspond to the lens powers in those directions, 

 /. e. to the maximum and minimum powers of the lens. If, 

 in comparing the curvatures of various arcs, a constant or 

 common sagitta is chosen, the curvatures are proportional to 

 the squares of the semi-chords of the arcs; and if the sagitta 

 of an arc is properly chosen, the curvature of that arc is 

 measured by the square of the semi-chord. It follows 

 therefore that the power of a lens or a surface in a given 



* Communicated by the Physical Society : read November 9, 1900. 

 f Phil. Mag. Mar. i900; Phys. Soc. Proc. 97, July 1900. 



