HOLLO AV SPHERES A.ND HOLLOW CYLINDERS. 215 



it is intelligible that the inner bright line, even with tolerably 

 high amplifications, will still fall in the marginal shadow of the 

 hollow space. 



To demonstrate the action of the four different systems of rays, 

 we have represented in Fig. 116 the distribution of light and 



FIG. 116. 



shadow for each separate path of the rays where R = 2 r. No. 1 

 shows the image of the twice-refracted rays which traverse the walls 

 in a straight line ; No. 2 represents the outer bright lines which 

 proceed from the rays which are twice refracted and once reflected 

 at the inner surface of the wall ; No. 3, the fine inner lines which 

 correspond to the rays which have entered the cavity and been 

 reflected once at the inner surface of the wall ; finally, No. 4 repre- 

 sents the marginal shadow which the rays traversing the lumen, 

 and consequently refracted four times, would of themselves pro- 

 duce. Since umbra and penumbra nearly coincide, we have through- 

 out, for the sake of simplicity, drawn only the limiting lines of the 

 umbra, and indicated for clearer determination the outer and inner 

 walls of the cylinder and the axis C C by lines through the figure. 

 If we regard the hollow cylinder, without reference to the plane 

 of adjustment, simply as a refracting apparatus, it acts upon the 

 marginal rays, which do not enter the lumen, as a sphere of equal 

 density ; but on the central rays, which traverse the hollow space, 

 as a concave lens. It forms, however, real and virtual images of 

 objects which are reflected in the mirror, both of which become 

 appreciable to the eye as soon as the instrument is focused to a 

 suitable level. The position of the real focus is, of course, approxi- 

 mately the same as with the sphere ; that of the virtual focus may 



