204 THEORY OF MICROSCOPIC OBSERVATION. 



By the help of these equations the limits of the umbra and 

 penumbra can easily be determined in any 

 given case. On account of the great difference 

 in the numerical relations which may occur in 

 practice, the discussion of a particular example 

 would be of no special value; a few of the 

 more general consequences may, however, be 

 noted. 



Let S R (Fig- HI) be a marginal ray of the 

 emergent cone of light, which (if produced 

 backwards) cuts the plane of adjustment (drawn 

 through C) in the point P. Then, as a glance at 

 FIG. ill. ^ ie fig ure shows, its angle of incidence a is equal 



to the angle E P C = 90 *|. If for this magnitude of the angle 



_i_ 5\ 5\ 



of incidence a a < 7- - or even less than - , then in 



4 4 



the former case the umbra will disappear, and in the latter the 

 penumtra. The sphere will therefore appear, under certain cir- 

 cumstances, uniformly illuminated from one margin to the other. 

 This occurs if its refractive index = 1-5, &> = 60, and 8 < 22 ; 



c\ 



whence a a 9J, and - j - ?; 9J. Globules of oil, 



spherical starch-grains, cylindrical hairs, &c., whose refractive 

 index does not differ much from that just assumed, will therefore 

 exhibit no marginal shadow with higher amplifications, while the 

 lower-power objectives show it the more distinctly the smaller 

 their angles of aperture. It would, of course, appear broadest 

 when observed with the naked eye, since must then be regarded 

 as infinitely small. 



If, when o> remains constant, the angle 8 varies, the distribution 

 of the light is altered as follows : If 8 = 0, that is, if the incident 

 rays are parallel, the equations for the umbra and penumbra will 

 be identical, i.e., the limits of both will coincide, or, what is equiva- 

 lent, the penumbra vanishes. If B gradually increases, the umbra 

 will become narrower and the penumbra broader. The inner limit 

 of the latter finally reaches the centre when 8 = &>, and then no 

 part of the sphere will appear so bright as the field of view. If 

 > o>, the limiting line of the penumbra will again move out- 



