of Lippiclis in Geometrical Optics. 115 



the tores. The cone may include a cylinder or plane as a 

 particular case. Suppose a telescope to sweep the horizon, 



Fig-. 2. 



rotating about a vertical axi?, then a vertical axial section o£ 

 its lens surfaces would trace a series of tores with the centres 

 of the generating circles in a plane, and a ray along the 

 optic axis of the telescope would be a common normal to 

 these tores. 



Lippich's essay is based on the principle of collinear 

 correspondence. This simple geometric relationship is, of 

 course, far from being satisfied for rays making finite angles 

 with the axis, but he remarks : — "The following properties 

 (developed in the first place for a single refracting surface) 

 of an infinitely thin bundle of rays become very simple, and 

 approximate very closely to the properties of paraxial bundles, 

 when the axis of the incident, and consequently the axis of 

 the refracted, bundle lies in a plane through the optic axis." 

 He then proceeds to establish a series of theorems, of which 

 the first few are identical with Young's well-known properties 

 of the aplanatic spheres. Homocentric pencils are first dealt 

 with, and then theorem No. 12, the one here discussed, paves 

 the way for the treatment of non -homocentric pencils. 



The proofs given by Lippich and Culmann involve the 

 assumption of certain properties of small pencils and Sturm's 

 focal lines. Owing to the elementary nature of the theorem, 

 the idea naturally suggests itself that it should be capable of 

 proof directly from first principles. Hence the following- 

 attempt, in which no optical assumption is made except the 



12 



