332 



Mr. R. F. Muirhead on 



Projjosition C. — All in-rays through a point a on the axis 

 intersect their out-rays in a fixed plane normal to the axis, 

 the base-plane for a. 



Fi-. £. 



Fiof. 9. 



Hence the following construction (iig. 9) for the image of 

 a given point P, having given two points a, j3 on the axis, 

 their images od and /3^, and their base planes. Draw P a, 

 P ^ meeting the base-planes of «, yS, in Ai' B/. Then P', the 

 intersection of a' Ki and /3' B/ is the image of P. 



Nodal Points, — If the base-plane of a is at infinity it is 

 clear that a and a' coincide with X and N', the first and 

 second nodal points of the system. 



Thus the property given in Proposition C may be looked 

 on as a generalization of the property of Nodal Points. 



We may note that if two axial object-points A and B with 

 their images A^ and B^ and their vertices a and /3 are given^ 

 as well as the planes of A, B, A' and B', then, as shown in 

 the last section, the out-ray for any given in-ray can be 

 constructed by drawing three straight lines, each joining a 

 pair of points, the third being the out-ray itself. 



On the other hand^ if A and B, their images A' and B', 

 and their base-planes a and h are given, we can constract the 

 image of any given object-point by drawing four straight 

 lines as shown in Proposition C. 



We may collect the results of our generalization as to the 

 cardinal points into a single figure as follows : — 



Let Aq be a point on the axis of which A is the image, 

 and let A^ be the image of A. Let E be the vertex for Aq, 

 and E' its image. Let B be the point for which A is vertex, 

 and B^ its image. 



Then if A goes to infinity Aq and A^ become the first and 

 second focal points, E and E' the first and second nodal 

 points, and B, B' the first and second principal points. 



