798 Transactions of the Soeietij. 



Though this proposition allows a comparison of those objectives 

 only which have equal apertures and are of similar construction, 

 it leads to important inferences. Firstly, it shows that the 

 characteristic constant quantity u, which is the " measure of per- 

 fection " of the objectives, does not require a separate determination 

 for every single system. If the value of u be known for one 

 system, it is known for all systems of the same kind, i. e. for all 

 which have the same aperture, are constructed on a similar formula 

 and with an equal degree of technical skill. Secondly, the pro- 

 position indicates the method by which a direct comparison of 

 objectives of different focal lengths may be obtained in regard to 

 the quality of images of equal amplification. 



Suppose the angular dissipation of the light — the constant angle 

 u — to be given for a particular kind of objectives of definite aperture. 

 If any one of these objectives has a focal length = /, its normal 



amplification is [N] = - (Z = 250 mm. or 10 inches). If now the 



total amplification of the Microscope is required = N, the necessary 

 super-amplification to which the said system must be subjected 

 will be 



_ N _N 

 " - [N] - / •^' 



and this super-amplification will introduce an angular dissipation 

 of the rays at the ultimate images, which is shown by the 

 expression 



Consequently : For objectives of the same kind and the same 

 aperture, but with difierent focal lengths, the manifestation of the 

 inherent defects and aberrations under a given power of the 

 Microscope is always in direct proportion to the focal length (or 

 in the inverse proportion of the objective-power) by which such 

 amplification is obtained. 



2. If we could suppose objectives of ideal perfection — absolutely 

 free from all technical defects and all unavoidable aberrations — the 



positions of the image, belong to tliat kind. Tlie proposition will therefore hold 

 good for all objectives of a similar formula and equal aperture. 



The assumption of a proportionate reduction of the geometrical defects 

 (defects of figure, centering, &c.) with decreasing focal length, which is implied 

 in the demonstration above, will not be in full accordance with the actual circum- 

 stances. In practice the relative accomplishment of smaller lenses will be inferior 

 to that of larger ones. According to the experience of the autiior, the difference 

 is, however, not very considerable except when the dimensions are very minute. 

 Though some difficulties of exact W(jikmanslup are increased with smaller dimen- 

 Bions, tiiere are others which are diiuinislied ; and the relative amount of those 

 defects which cannot be overcome by careful ami skilful work, may therefore be 

 considered as nearly equal for ull focal lengths, within rather wide limits. 



