The Relation of Aperture and Poiver. By Prof. E. Ahhe. 793 



strictly residuary aberration as well as the irregular dissipation of 

 the rays in consequence of technical faults of the lenses) in the 

 ultimate image of the entire Microscope is, with every given objec- 

 tive, always proportional to the total amplification of the image, 

 and does not depend on the length of the tube alone, or the depth 

 of the ocular alone, with which that amplification may be obtained. 

 This is easily seen, if it is borne in mind that the ocular merely 

 effects an enlargement of the objective-image, together with the 

 dissipation-circles which are inherent therein. For if a certain 

 total amplification N — say 500 diameters — is obtained with the 

 whole Microscope, the objective amplifying the object by W 

 diameters, and the ocular amplifying the objective-image by N'' 

 (say 50 and 10 respectively), then will N'N" = N, and the hnear 

 diameter of the dissipation-circles in the ultimate image will be 

 N" . e, if 6 denote the diameter of the dissipation-circles in the 

 objective-image. If now the same total amplification N should 

 be obtained with the same objective by means of a longer tube 

 and a lower eye-piece, W will be increased (say to 100), and in 

 the same proportion e also, but N" will be diminished in the 

 inverse ratio (to 5). The product N" e therefore retains its 

 former value. But if, on the other hand, the total amplification 

 N should be increased (either by increasing the length of the tube 

 and therefore the value of N', or by increasing the amplification 

 of the ocular IS"), the product We will vary in the ratio of N, 

 because in the one case the second factor, and in the other case the 

 first factor, are increased in that ratio. 



(h) According to a fundamental dioptrical proposition, the 

 linear amplification N' of the image, which is projected by a system 

 of given focal length/, is strictly determined by the formula 



in which A denotes the distance of the image from the posterior 

 principal focus of the system (the place where rays are collected 

 from distant points in front of the system) ; and this is the same 

 whether the image be real or virtual. The objective-image of a 

 given system is therefore always amplified in exact proportion to 

 the length A; and the linear diameter of the dissipation-circles 

 (e) of that image must also be proportional to A, since e is propor- 

 tional to W. Taking now the angular diameter of these dissipa- 

 tion-circles at the posterior principal focus, i. e. the visual angle 

 under which they would appear at that place, this angle must 

 obviously be the same for every position and amplification of the 

 image, because the linear diameter e always varies in direct propor- 

 tion to the distance A. We thus arrive at the theorem : — 



If an objective projects a real or virtual image without the 

 Ser. 2.— Vol. III. 3 H 



