HOLLOW SPHERES AND HOLLOW CYLINDERS. 



1-5 



209 



, and R as unity. The quantities F and 



index is taken as 



r are therefore expressed in fractions of half the longer diameter. 



The virtual focus is therefore always further from the centre 

 than the inner bounding surface of the cylinder-wall. Since then, 

 as the calculation shows, all rays, whose angles of incidence are 

 with equal radii somewhat larger or smaller than those above 

 denoted, invariably appear to come from points which lie nearer to 

 the periphery, it follows that the innermost part of the wall will 

 fall for a breadth of F r in the umbra, while the peripheral part 

 receives (just as the margin outside the rings in the air-bubble) 

 a faint illumination, which is, however, intensified under the 

 above conditions by the marginal rays as mentioned in (1). If 

 the surrounding medium is water, and if co and 8 are taken toler- 

 ably large, the resulting brightness will therefore in many cases 

 equal that of the field of view, and thus the bright line will appear 

 in the direction of the margin without definite limits, only slightly 

 increasing the effect of light. 



The assumption, usually tacitly made in practice, that the 

 lumen of a cylindrical cell reaches as far as the marginal shadow,. 

 is therefore, from what we have adduced, essentially incorrect. 

 The error is the greater in proportion to the refractive power of the 

 substance ; it is infinitesimal only when the object lies in a 

 medium of approximately equal density. This, of course, holds, 

 good also for the cavities in starch-grains, nuclei, globules of oil,. 

 <fcc. they all appear somewhat larger than they actually are. 



P 



