HOLLOW SPHERES AND HOLLOW CYLINDERS. 



207 



depends not merely upon the angles of aperture o> and S, but 

 also upon the thickness of the walls, whether and how much 

 they contribute to the illumination of the latter. If r is the 

 short and R the long radius of the hollow cylinder, the thickness 

 of the wall being therefore E - r y and 8 T L K (Fig. 112) is 

 a limiting ray which forms a tangent to the inner wall after the 

 first refraction, and if a, and a are its angles of incidence and 

 refraction, and n the refractive index ; we get sin a = r, sin 



. 

 For the marginal 



n sin a , . nr 



a = ^ , and consequently sin a = -^. 



nr 



rays the condition sin a > will therefore in general hold good ; 



since, moreover, if they are to contribute to the illumination of the 

 walls with medium focal adjustment, they must satisfy the general 



equation a a 



_i_ 



, they will be lost in the microscopic image 



as soon as -=- reaches a certain magnitude. They come into account 



therefore with the given refractive indices only with thicker walls. 



If, for instance, n = . onr (the refractive ratio between flint- 



I'OOO 



glass and water), and ^ = '8, we find a > 81, a i 53 8 7 , a -a' 

 A 



is therefore nearly 28 at the mini- 

 mum. The sum of the angles of 

 aperture co + S would therefore 

 amount to approximately 4 x 28 

 = 112, if the innermost rays only 

 contributed to the illumination of the 

 cylinder-wall. If no other rays illumi- 

 nated this cylinder- wall it would appear 

 perfectly black even with a somewhat 

 high amplification (ordinary illumina- 

 tion being assumed). 



The marginal rays mentioned in 

 (2), which at the inner surface of the 

 walls are reflected to the outer walls, 

 and there undergo the second refrac- 

 tion, form somewhat above the centre 

 a virtual focus, which as in the air-bubble nearly coincides with 



