348 Transactions of the Society. 



IX. — Note on the Proper Definition of the Amplifying Power of a 

 Lens or Lens-system, 



By Prof. E. Abbe, Hon. F.E.M.S.f 



(_Read I2th March, 1884.) 



The generally adopted notion of " linear amplification at a cer- 

 tain distance" is in fact a very awkward and irrational way of 

 defining the "amplifying power" of a lens or a lens-system. 

 Unfortunately, there is little hope that a more rational expression 

 will be generally adopted, because it will seem to be " too abstract," 

 but it may, nevertheless, be useful to consider the following : — 



In the formula N = - the " amplification" of one and the 



f 

 same system varies with the length Z, or the " distance of vision," 



and an arbitrary conventional value of I (e. g. 10 in. or 250 mm.) 

 must be introduced, in order to obtain comparable figures. The 

 actual " linear amplification" of a system is, of course, different, in 

 the case of a short-sighted eye, which projects the image at a dis- 

 tance of 100 mm., and a long-sighted one which projects it at 

 1000 mm. Nevertheless, the " amplifying power" of every system 

 is always the same for both, because the short-sighted and the long- 

 sighted observers obtain the image of the same ohject under the 

 same visual angle, and consequently the same real diameter of the 

 retinal image. That this is so will be seen from fig. 48, where 

 the thick lines show the course of the rays for a short-sighted 

 eye, and the thin lines for a long-sighted one, the eye in each 

 case being supposed at the posterior principal focus of the system. 



The semi- visual angle u* under which an object of semi- 

 diameter h is seen, is the same for both observers, as the change 

 resulting from the different positions of the object concerns 

 only the degree of divergence of the various pencils from the 

 various points of the object (and the image), and does not alter the 

 refraction of the principal (central) rays from the various points. 



This consideration leads to an expression of the " power" which 

 is in conformity to the last-mentioned salient fact. The quotient 



tan u* 



where u* is the semi-visual angle corresponding to h the semi- 

 diameter of the object, is a constant quantity for every system, not 

 depending on the particular circumstances of the observing eye ; 

 and this quotient indicates, obviously, the greater or smaller visual 



t The original paper is written by Prof. Abbe in English. 



