334 Mr. K. F. Muirhead on 



Hence M N' is parallel to C X. Hence Q', M, N' are col- 

 linear. 



Thus the oiit-ray for Fj Q is R Q^ and the out-ray for Q N is 

 Q^ N'. Hence Q' is the image of Q. Hence all points in Q D 

 have their images in the same line, and D is its own image. 



Having proved the existence of two Double-Points, which 

 we shall name Di and Dg, we may proceed as follows : — 



If Q be any point in a double-plane its image Q^ may be 

 got by drawing N^ Q^ parallel to N Q to meet the double- 

 plane in Q^ 



It follows that D Q' : D Q = D N' : D N, a ratio independent 

 of the position of Q in the plane of D. 



This of course is a special case of the theorem that a set 

 of points in a plane perpendicular to the axis form a figure 

 similar to its own image. 



The general construction for the image of a given object- 

 point when the axis and two extra-axial object-points with 

 their images are given, requires modification when a given 

 point and image have the sajne plane, i. e. when they lie in a 

 double-plane. The vertex of the point then coincides with 

 the double-point. But the constancy of the ratio D Q^ : D Q 

 gives an obvious construction. If P and P' be the given 

 point and image in the plane of D, and the image of Q (also 

 in D) be required, we have onlv to make DQ^:DQ = 

 D P^ D P. 



Again, taking 



/=FiH=N^F/,/=FiN = H'F/, A=HH'=NN^ 

 .;=FiD, 

 we have, since 



FiD:FiH = DQ:HE = DQ: DQ'=DX: BW 



X F,N-.2? f-x 



Hence the midpoint between D^ and Do also bisects NH^, 

 HN^andF/F,'. 



Let this point be called 0. It may be named the midpoint 

 of the dioptric system. 

 ' We have Dr'rzzO D/ = F^-F H . FN. 



This affords another construction for Dj and D2 when the 

 cardinal points are given. 



Bisect N H^ in and draw L perpendicular to the axis 

 having its length a mean proportional between / and f\ 

 AYith L as centre and F^ as radius, describe a circle which 

 will cut the axis in D^ and Do. 



