GLOBULES OF OIL IN WATER. 205 



wards, and similarly that of the umbra; and both will finally reach 

 the upper surface of the sphere, depending on the magnitude of co. 



If these theoretical deductions had to be somewhat modified 

 by the inequality of the loss, which the differently-inclined rays 

 undergo in consequence of the reflexion at the upper surface of 

 the lens, as well as by the aberrations of the object and of the 

 Microscope, yet the experiment teaches that the calculated limits 

 between light and shadow for every combination of o> and will 

 agree approximately with those observed. If, for instance, with 

 the ordinary illumination, we observe through the body-tube (after 

 the removal of the objective and eye-piece) a glass sphere or a 

 cylindrical glass rod in the incident cone of light, the illuminated 

 middle part will appear almost as a point or as a line, and the 

 broad marginal shadow deep black. If, however, we hold the same 

 objects against the open sky, that is, where S is very large, they 

 will appear bright from one margin to the other. 



The focal length / of a sphere or a cylinder is determined for 

 parallel rays by the formula 



ri T 



J := 2iri"^~^j 9 



where ri and n denote, respectively, the refractive indices of the 

 sphere and the surrounding medium, and r is the radius of the. 

 sphere. If, for instance, ri = 1*5 and n Q = 1, then the distance 

 of the focal point from the centre is f r\ hence, if we adjust the 

 Microscope to the corresponding level, the real image of more 

 distant objects, from which direct light reaches the object through 

 the diaphragm, will be seen there. This is exhibited also by 

 globules of oil and spherical starch-grains, as well as by cylindrical 

 hairs, bast-cells, &c., with striking clearness by withdrawing the 

 diaphragm some distance, or if the incident rays are otherwise 

 rendered parallel. Under favourable circumstances, it is even 

 possible, from the known distance and size of the object, and the 

 accurate measurement of the image, to determine to the second 

 decimal place the refractive index of the substance. The formula 

 to be applied in this case can easily be developed from the one just 

 found for the focal length, 1 and the known relation of the conjugate 



1 The formula for the focal length needs, moreover, a slight correction on 

 account of the spherical aberration of the marginal rays. Our observations 

 regarding the focal length of the air-bubble (vide note on p. 201) are applic- 

 able here also. 



