;i6 



SUMMARY OF CUKKENT RESEARCHES RELATING TO 



It follows, therefore, that if the circle be suitably placed, the angle 

 of convex reflection at the circle will be equal to the angle of radius 

 vector rotation. The figure assumes that the proper position of the 



Fig. 71. 



Fig. 72. 



circle has been found : it only remains to fix the position of M, its 

 centre. 



In the isosceles triangle B P' C- 



Again, 



2 B C cos u = R = /' (1 + cos u). 



B C = B M + M C = - sec u + M C 



•> 



whence by substitution and reduction M C = -. The point C, being the 



cardioid-cusp, is fixed, and the circle centre is therefore known. It will 

 be noticed that the construction does not depend upon the value of r. 

 which may consequently be chosen to suit the lens-maker's convenience. 



The radius r may also be expressed by r = , and this property 



makes the ray aplanatic in Abbe's sense of the word. 



Fig. 72 shows how the construction is applied by Messrs. Zeiss on 

 their aplanatic or cardioid condenser for dark-ground illumination. 

 Parallel to the axis incident rays pass through a ring-shaped opening in 

 the central diaphragm of the condenser. The diaphragm is set accurately 

 perpendicular to the condenser axis. The shaded part of the figure re- 

 presents solid glass, and the course of the rays and their emergence and 



