of Lippich's in Geometrical Optics. 125 



the incidence; while if LN is small and ON finite, LO must 

 be finite, and therefore by (ii.) w small of first order. Hence, 

 it' the second plane of incidence be brought into coincidence 

 with the first by rotating about L the line of intersection, the 

 points C, D, N will coincide with C, D', N' to the first 

 order. 



It may be noted that if co is finite, the angle of incidence 

 must be small, since by (i.) and (ii.) both LN and LC, and 

 therefore ON, must be small. But the converse is not true. 

 It is clear that if C coincides with N the planes of inci- 

 dence may have any inclination without contravening the 

 hypothesis. 



Projection on the Sagittal Sections. 



Let P" be the projection of P on the first sagittal section, 

 and t the line of intersection of the two sagittal sections. 

 t is perpendicular to the tangential section, and is a tangent 

 to the surface along a principal section. 



To show that the projections meet in a point we may 

 proceed similarly to Culmann : — 



The projection of any ray through P must (I) pass 

 through P", and (2) intersect the line t. Let it intersect 

 at R (see fig. 5, where the straight line QR or t is supposed 

 at right angles to the chief plane of incidence). 



Fisr. 



Let the plane through P parallel to the tangential section 

 (chief plane of incidence) cut t at Q. 



P"Q is parallel to the chief ray, since the sagittal plane 

 cuts two parallel planes in parallel lines. By hypothesis the 

 ray incident at P is nearly parallel to the chief ray, hence 

 the angle QP"R is small. "Also P"Q is given small. There- 

 fore QR is small of the second order, and R may be taken 

 as coinciding with Q. Similarly the projection of the re- 

 fracted ray on the other sagittal plane passes through Q. 

 Q being on the tangent plane at 0, the point of incidence 

 of the chief ray, and near 0, may be regarded as a point 

 on the surface. Hence the projections of the incident and 

 refracted rays meet at a point of the surface. 



