Lippich's Projective Theorem in Geometrical Optics. 113 



surface is bounded by any curves and the receiving surface 



is bounded by a circle provided the same conditions are 

 satisfied. 



Any further attempt to generalize by allowing the bounding 

 curves of the receiving surface to be non-planar reduces at 

 once to the original theorem, since a non-planar curve cannot 

 lie on more than one sphere. 



Research Laboratories of the 

 General Electric Co., Ltd., London. 



X. On a Projective Theorem of Lippiclis in Geometrical 

 Optics. {With a Note on the Equations of the Projection 

 of a Straight Line on a Plane.) By Alice Everett *. 



THE theorem referred to is the following : — If the corre- 

 sponding incident and refracted portions of a ray, 

 which is infinitely near to a chief ray lying in a principal 

 section of a refracting surface, be projected upon either the 

 tangential or sagittal sections of the chief ray, then the 

 projections also correspond. 



By near rays are meant rays which are nearly parallel 

 and are incident at near points where the normals are nearly, 

 parallel. 



The tangential section is defined as that principal section 

 of the surface, at the point of incidence of the chief ray, 

 which contains the chief ray and normal ; it coincides with 

 the plane of incidence. The sagittal plane is defined as a 

 plane through the chief ray perpendicular to the plane of 

 incidence. The sagittal sections for the incident and re- 

 fracted chief rays are in general different, and not principal 

 sections. 



The theorem was proved by Lippich (1877) for the sphere 

 only, in an essay on Refraction and Reflexion of Infinitely 

 Thin Ray Systems by Spherical Surfaces (Vienna Academy, 

 Denhchriften, Band 38, p. 17 G (1878)). Culmann extends 

 it to non-spherical surfaces in his chapter in von Rohr's 

 ' Theorie der optischen Instrumente,' Band 1, pp. 183-185. 



Its interest lies in the fact that it enables the path, after 

 refraction at a non-spherical surface, of any oblique ray 

 infinitely near to a ray in a principal section to be found by 

 applying to its projections the ordinary method adopted for 

 rays in an axial plane of a spherical refracting surface. We 



* Communicated by the Author. 

 Phil. Mag. S. 6. Vol. 40. No. 235. July 1920. I 



