DETERMINATION OF THE CARDINAL POINTS. 



187 



(column " Observed "), which we give for comparison with the 

 values calculated according to the above formulae. 1 



The most striking peculiarity of the objective examined, and to 

 which we draw special attention, is the fact that the anterior lens 

 is stronger than the whole system. The equation of condition for 

 this case is an immediate result from the formula for the combined 

 focal length, taking 



* 



and then dividing both sides by < , we obtain 



Of the medium-power systems we have examined, the No. 9 of 

 Beneche is the only one possessing this peculiarity ; neither of the 

 corresponding objectives of Oberhreuser or Plcessl has it. On the 

 other hand, it occurs somewhat frequently in low-powers, 

 especially those composed of two double-lenses separated con- 

 siderably. We observed it, for instance, in the No. 4 of Beneche, 

 and the No. 4 of Hartnack. In the No. 9 immersion of Hart- 

 nack, the anterior lens is also somewhat stronger than the whole 

 combination ; and this may possibly be the case in the highest 



1 The differences in this table between the observed and calculated values 

 might be explained by the fact that the determination of <7 2 is impossible for 

 the single lenses with sufficient accuracy, on account of the indistinctness of 

 the image in the corresponding position. Moreover, the supposition of equal 

 focal lengths in both positions is, strictly speaking, only justifiable for central 

 rays, and becomes untenable for marginal ones, because the amount of the 

 spherical aberration, and consequently also the distance of the focus on 

 reversal of the lenses, is slightly altered. Such an alteration of distance is 

 noticeable even in deeper lenses as soon as the conjugate foci are in essentially 

 different ratio. For this reason the values, for instance, of the focal lengths 

 found according to different methods, never exactly coincide. 



