1895.] in Geometrical Optics. 311 



and those of the refracted pencil normals to the surface 



the origin being on the interface, the axis of z along the axis of 

 the corresponding pencil and the axis of x in the plane of refraction, 

 then A^ is connected with A^ and B^ with B^ by equations of the 

 same form as in the special case of § 6, while Cj is connected with 

 C2 by a similar equation. If lengths equal to A-^ and A^ are laid 

 off along the axes of the pencil, the line connecting their extremi- 

 ties passes through a fixed point ; a similar statement applies to 

 the other two pairs of points, and the coordinates of the three fixed 

 points thus obtained have been given by Maxwell. The graphical 

 solution may now be completed by assigning geometrical con- 

 structions for them. The fixed point connected with A^ and Ao, is 

 determined by Young's construction (§ 4) in which the circle is 

 the circle of curvature of the section of the interface by the plane 

 of refraction. The fixed point connected with B^ and B^ is the 

 centre of curvature of the normal section perpendicular to the 

 plane of refraction. 



The connexion between Cj and C^_ is^ through the equation 

 cot ^1 cot 02 _ cot 01 — cot 02 



where 0i and 02 are the angles of incidence and refraction, and S~'^ 

 is the coefficient of the product term in the equation of the inter- 

 face. Thus, when G^ is infinite, Cg is equal to 



8 sin 01 cos 02/sin (0i — 02) ; 



and when C^ is infinite, C^ is equal to 



— 8 cos 01 sin 02/sin (0i — 02). 



Hence, draw round the point of incidence a circle of radius 8, 

 in the plane of incidence : from the point where each axis meets it 

 draw a perpendicular to the normal at the interface, and from its 

 foot a parallel to the other axis of the pencil : these parallels meet 

 in the fixed point required, which is in fact also on the line connect- 

 ing the points in which the two axes meet the circle. 



8. Any system of negligible thickness. It follows (§ 3) from 

 the elements of the theory of perspective, that when a general 

 pencil crosses any optical system whose thickness is negligible, the 

 refracted pencil may still be deduced from the incident one by the 

 construction of § 7 ; except that the fixed points must now be 

 determined either once for all experimentally, or else by a succes- 

 sion of linear constructions, one for each refraction. 



1 Maxwell, Proc. Lond. Math. Soc, iv. 1872 ; vi. 1874. 



