IMPORTANCE OF THE ANGLE OF APERTURE. 93 



2. IMPORTANCE OF THE ANGLE OF APERTURE. 



Since the incident pencils are not dependent upon the angle of 

 aperture of the objective, we have still to inquire how the magni- 

 tude of this angle influences the microscopic image. It might be 

 supposed that the action of the larger aperture is dependent upon 

 a favourable distribution of light and shadow, because globules of 

 oil, starch-grains, &c., exhibit a broad marginal shadow with a 

 moderate amplification, which disappears if a more powerful 

 objective with larger aperture is used; or that through an extension of 

 the luminous parts an equalization of shadow and light might, to a 

 certain extent, be established, in the sense that with striated objects, 

 for instance, the lines of shadow and light would be equalized in 

 breadth, and hence more easily distinguishable. Such an explana- 

 tion would be untenable, for, in the first place, the luminous parts 

 of many test-objects preponderate so much in the image that a 

 diminution would be advantageous ; but yet their resolution is only 

 possible by means of objectives of large angular aperture, there- 

 fore the distribution of light and shadow depends not only upon 

 the objective, but also upon the size of the diaphragm, as will be 

 more fully shown later on. The angle of deflexion of the rays, 

 which corresponds to the limiting points of the inner shadow and 

 the penumbra, is always determined by formulae in which the two 

 angles of aperture &> and 8 enter only as the sum or difference ; 

 and though the limiting points themselves are not thereby mathe- 

 matically given, their position is nearly the same as if the refraction 

 had taken place in an equivalent infinitely-thin lens in the plane of 

 the object. It is, at any rate, permissible in our discussion to 

 substitute such a lens for the refracting surface, such as a diatom- 

 frustule, without influencing the accuracy of our conclusions. It 

 is, moreover, quite immaterial in theory whether an objective of 60 

 aperture be combined with a diaphragm of 30, or, conversely, an 

 objective of 30 with a diaphragm of 60, the difference in the 

 image will be hardly appreciable. We will demonstrate this by a 

 diagram. 



Let a b (Fig. 42) be the equivalent lens, c its centre, and//,/'/' 

 its cardinal focal planes on either side, which consequently inter- 



