the A.vial /dioptric tS/fdeni. 335 



Section III. 



In Section I. wo showed that when the axis and two extra- 

 axial points with their images are given we can in general 

 determine the image o£ any point, and the cardinal points oi" 

 the system. The question arises as to whether any necessary 

 relation exists between four such points and the axis, or 

 whether we can always find a dioptric system for which a 

 line and four points taken at random are axis, object- and 

 image-points. The latter alternative would be proved true 

 if we could show that the fundamental construction given in 

 Section I. gives a transformation of object-points into image- 

 points in such a way that any two pairs of these being taken 

 for the given points, the same transformation as before would 

 be arrived at. For in that case we should know that, starting 

 from four cardinal points, the same transformation is arrived 

 at, and it is easy to show that by means of three media and 

 two refracting surfaces we can form a dioptric system having 

 any four arbitrarily chosen cardinal points on an axis. 



Let CA, CA', C B, CB' be four lines concurrent in C, 

 and a, /3 two fixed points. Let P Q, any line in the plane 

 of the figure, cut C A in P, C B in Q. Let P cc meet C A' in 

 P^ and Q yg meet C B^ in Q^ 



Thus the position of the line P^ Q^ is uniquely determined 

 by that of the line P Q. To denote this relationship we shall 

 call P^ Q' the iieio line corresponding to the old line P Q. 



The only old lines in the plane for which the construction 

 fails to determine the new lines are those passing through C. 

 For such lines the points P Q P^ Q^ all coincide with C, so 

 that the new line is only partially determinate. We might 

 by applying the method of limits extend our definition so as 

 to make such new lines fully determinate, but we prefer to 

 complete the theory without having recourse to that method. 

 We proceed to prove that (subject to the exception above 

 mentioned) all old lines through a given point have new 

 lines which are concurrent. 



Let L Po Qo. L Pi Qi, L P2 Q2 be three straight lines 

 through L cutting C A in Pq, Pi, P2 and C B in Qq, Qi, Q2. 

 Draw P, «Pc', Pi« Pf, ^2 « W meeting C A^ in P-j', P/, P2^ 

 and draw QuySQo'^ Qi/SQi'; Q2i3Q2' meeting C B^ in Qo', 

 Q/, Qj. Denoting by 0{ABCD|- the anharmonic ratio of 

 the pencil A, B, C, D, we have 



«{C Po^ P/ P/H^-lGPo P, P2[ =L^C Po Pi P2[ 



= L{CQoQiQ2}=;8{CQoQiQ2}=/e{CQo^Q/Q2'}. 

 Thus the equi-anharmonic ranges Pq' Pi' P2' and 



