The Relation of Aperture and Power. By Prof. E. Ahbe. 797 



if an amplification of 200 is required with a l-inch instead of a 

 1-2 inch as in the preceding example, the necessary super- amplifica- 

 tion will be = 20, and all aberrations and other defects inherent to 

 the system will appear in the image under twice the visual angle 

 of that in the other case. 



(d) In order to compare the performance of different objectives 

 under various powers, it will be necessary and sufficient, according* 

 to the foregoing theorems, to determine for any given system the 

 constant quantity u, by which the inherent dissipation of the rays 

 is measured. 



One part of this problem may be settled- by means of the 

 following proposition : — With objectives of equal aperture, similar 

 construction, and equal degrees of technical excellence, the constant 

 visual angle of the dissipation-circles is always the same and inde- 

 pendent of the focal length. 



This may be proved by a very simple consideration. Suppose 

 a system A of a certain aperture and given focal length / to be 

 brought to the best possible correction of which the construction 

 may admit for a certain distance of the image. Another system B 

 of exactly similar composition may now be obtained by reducing 

 the linear measures of all the elements (all radii, diameters, 

 distances, &c.) and all technical defects of figure and positions of 

 the lenses in the same proportion (say, e.g., of 2 : 1), just as if 

 the diagram of the system and the transmitted rays had been 

 drawn on a reduced scale. The focal length will thereby be 

 changed in the same proportion (/ : J/) and also the distances of 

 the conjugate foci of best correction ; but the aperture will not be 

 changed, and the angles of all emerging rays — of regular or 

 irregular transmission — will be the same as the angles of the 

 corresponding rays in A, by virtue of the strict geometrical simi- 

 larity of all the elements. If now the space over which the rays of 

 one pencil are dissipated at the image of A, subtends a certain 

 angle u in regard to the posterior principal focus of A, the same 

 angle u must obtain for the image of B in regard to the corre- 

 sponding principal focus of B ; and that angle u must persist, as 

 has been shown, if B should afterwards project an image to any 

 other distance (e, g. at a corresponding distance to A), provided 

 the best correction for the new position of the conjugate foci be 

 obtained. Consequently the angular value of the dissipation-circles 

 (m) is the same for all similar systems, however difierent the focal 

 lengths may be.* 



* Strict similarity cannot of course obtain, except when the distances of the 

 conjugate foci, for which the objective is corrected, are proportional to the focal 

 length. We may, however, disregard all differences of construction which could 

 be effected, or undone, without introducing new essential aberrations ; and those 

 changes of a system which are necessary in order to compensate for different 



