I 



the A.vial IHoptric System. 33.^ 



Douhle Points and Planes, oy Brava'is Points and Planes. 



It was Braviiis who pointed out in 1851 the existence of 

 two points on the axis, each of which is its own image. The 

 easiest method of determining these points is probably by 

 expressing algebraically the relation between the abscissae of 

 an axial point and its image (which can easily be deduced 

 from our fundamental construction) and then equating these 

 abscissae, which gives a quadratic equation for the abscissa 

 of a double point. But for the sake of uniformity of treat- 

 ment we shall investigate the double points without using 

 algebra (lig. 10). 



Let Fi, Fg', H, H', N, N' have the same significations as 

 before. Take B on the axis so that X B = H^ H. Let Fi R 

 be any ray through Fj, meeting the plane of H in R. 



Let a line parallel to the axis, equally distant from H and 

 R, meet Fi K in U, and the plane of B in V. 



Through N draw^ a line meeting Fj E. in Q, U V in Gc, and 

 BY in C, such that CG = GQ. (In Thomas Simpson^s 

 ' Geometry,' a construction is given for a problem including 

 this as a particular case, It has tw^o solutions, real or 

 imaginary.) 



Draw Q Q' D perpendicular to the axis, meeting the axis 

 in D and a line through R parallel to the axis in Q'. D is a 

 double point. 



For QQ' = BC. Hence if we complete the parallelogram 

 Q' Q X M, we have N M = C B. 



Hence the triangles C B N^ M N X' are equal in all respects. 



