ZOOLOGY AND BOTANY, MICROSCOPY, ETC. 323 



Thus the divergence or convergence of the emergent pencil is 

 completely defined by N, u, and n, without requiring any knowledge 

 of the focal length of the system, or of the distance d at which the 

 image is formed. 



Now there is of course loss of aperture (1) when there is a loss of 

 amplification N, while u* remains the same, and (2) when there is a 

 reduction of u*, while N remains the same. For these reasons : — 

 With any given distance d of the image from the back lens, d tan u* 

 is the clear available semi-diameter of the back lens. If now the 

 objective gives less amplification (at the same distance d) while u* 

 is not greater, we should have a loioer-poioer objective with the 

 same clear diameter of the back lens, and this is necessarily loss of 

 aperture. If, on the other hand, the system gives a narrower pencil 

 (diminished u*) while N is not greater, we should have an objective 

 of the same power giving a narrower emergent pencil (i.e. with a 

 smaller clear diameter of the back lens), and this is necessarily loss 

 of aperture also. 



Therefore, constant aperture requires the condition of constant 

 amplification N for the same distance d (i. e. for the same length of 

 tube) if u* is the same, and of constant angle u* of the emergent 

 pencil if N is the same. It follows, therefore, that the remaining 

 element in the formula which relates to the anterior pencil (J^ sin u) 

 must also be constant ; so that there is always loss of aperture 

 whenever the product n sin u has a smaller value, as this would 

 require either a smaller N or a smaller u*. 



If now in an immersion objective, with balsam or oil in front, 

 u is greater than the critical angle of the medium, n sin u will be 

 > 1 (for n is 1*5 and sin u is at least '667). It will be impossible 

 to obtain the same value, if the front medium is changed for air, for 

 n being then only = 1, sin u must be greater than 1, that is an angle 

 with a sine > 1, which is absurd ! 



No alteration of the optical system is of any avail because the 

 formula holds good for every system. 



Therefore no dry objective can be equal in aperture to a wide- 

 angled immersion objective in which n sin m is > 1 — i. e. the balsam 

 angle of which exceeds 82°. 



It follows also that equal or different apertures always require 

 equal or different values of the expression n sin u, which is therefore 

 the proper expression for aperture in general. 



pointed out its connection with the essence of aplanatism. He indicated at the 

 same time a simple experimental demonstration of the law. 



For the literature on the subject, see the following : — 



Abbe, " Beitrage z. Theorie d. Mikroskops," &c., Arch. f. Mikr. Anat., ix. 

 (1873) p. 420. 



Helmholtz, " Die theoretische Grenze fiir die Leistungsf ahigkeit des Mikro- 

 skops," Poggendorff's Annal. Jubelband (1874) p. 566. 



Abbe, " Ueber d. Bedingungen d. Aplanatismus d. Linsensysteme," ' Carl's 

 Repertorium fiir Experimentalphysik,' xvi. p. 303. Conf. this Journal, iii. (1880) 

 p. .509. 



The formula in question (the outcome of the combination of the Lagrange- 

 Helmholtz law with the law of aplanatic convergence) is the basis of all investi- 

 gations on dioptrical questions wliich deal with wide angles. 



