the A.ridl I t'loptric System. 



331 



us to construct the in 



of anv object: the axis, one extra- 



axial point A with its image A', anil two other tLvlal points 

 K. L with their images K' L' (fig. 7). 



Fig. 7. 



The intersection of A A' with the axis gives a. the vertex 

 for A. Let P be any object-point. Draw K P Q meeting 

 the plane of A in Q. Draw Q ai Q' meeting the plane of 

 A' in Q'. Then K' Q' contains the image of P. 



Draw L R P cutting the plane of A in R, and R a R' 

 cutting the plane of A' in R'. Then L'R' contains the 

 image of P. 



Hence P' the intersection of K' Q' and L' R' i- the required 

 image. 



Xow if we send a, to infinity A and A' lie in the first and 

 second principal planes. If w^e send K and L' to infinity, 

 L and K' become the first and second focal points. 



The ordinary construction for the image of a point bv the 

 aid of focal points and principal points is obviously a particular 

 case of that just given. 



Section II. 

 Correlative Fundamental Theory, 



Corresponding to the whole of the theory of the first 

 section based on Proposition B there can be developed 

 another having a dualistic relation to it, based on the theorem 

 that all in-rays through an axial point meet their correspond- 

 ing out-rays in a certain plane normal to the axis. 



Let AP (fig. 8) be a line perpendicular to the axis A A', 

 and A' P' its image, and let P P' cut the axis in ex,, so that a. 

 is the vertex for A. Let a' be the image for a. Then the 

 in-ray at, P meets its out-ray u' V in V. So long as P is in 

 the plane of A the positions of a and of A' are fixed. The 

 plane of A^ we shall call the hase-plane for a, or for any in- 

 ray passing through a. Thus we have : — 



