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Miss A. Everett on a Projective Theorem 



natural law of refraction The question being really one of 

 differential geometry, this method has been chosen as the 

 most suitable to apply, though another very elementary 

 proof by spherical projection is appended. 



The analysis brings to light the following facts overlooked 

 by Lippich. In the case of the tangential projection, the 

 theorem has a wider application than he assigned to it, for here 

 the variable ray need not be nearly parallel to the chief ray 

 itself, but only to its plane of incidence. On the other hand, 

 in the case of the sagittal projection, small quantities of the 

 first order have to be neglected unless the angle of incidence 

 is small, whereas in the tangential projection only small 

 quantities of the second order need be neglected. Thus the 

 theorem holds less accurately for the sagittal than the tan- 

 gential projection. 



When the chief ray is incident normally, the sagittal plane 

 is a principal plane, and the two cases are interchangeable. 



It seems improbable that a theorem of this nature should 

 have been left undiscovered till Lippich's time. The writer, 

 however, has so far been unable to find any mention of it by 

 earlier investigators, and would be obliged for references. 



Proof. 



1. Projection on the Tangential Section. 



Fisr. 3. 



The condition that the plane of projection shall be a 

 )rincipal section is clearly necessary to ensure that the 



