of Lippiclis in Geometrical Optics. 123 



Elementary Proof. 



The Hollowing is an attempt at an Elementary Geometrical 

 Proof of Lippich's Theorem. 



In order that two rays may be conjugate : — 



(1) They must meec in a point on the refracting surface. 



(2) Their plane must contain the normal to the re- 



fracting surface at that point. 



(3) The angles 0, <£' which they make with the normal 



must satisfy the relation // sin <$>' = p sin 0, where 

 /x, fx' are the refractive indices. 



Projection on the Tangential Section. 



Let be the point of incidence of the chief ray. If V 

 be the projection of P, the point of incidence of a neigh- 

 bouring oblique ray, upon the plane of incidence of the chief 

 ray, it is well known that the distance of P' from the tangent 

 plane at (the point of incidence of the chief ray) is of the 

 second order of small quantities. Hence we may regard 

 P' as lying on the surface, and the projections of rays and 

 normal through P pass through P'. This disposes of con- 

 dition (1). Condition (2) is satisfied if the plane of 

 projection is in a principal section at 0. 



Suppose a sphere described about as centre (figs. 4 

 and 6). 



Let the (chief) ray incident at cut the sphere at A, 



,, „ ,, ray refracted at ,, ,, „ „ B, 



„ ,, normal at 

 a radius parallel to the ray incident at P 

 „ ,, ,, „ ray refracted at P 



,, ,, normal at P 



15 55 



55 



„M, 



c 



55 55 



55 



5? ^J 



D 



55 55 



55 



'? *-* i 



V )l 



y> 



„N. 



