Some Remarks on the Apertometer. By Prof. E. Ahhe. 27 



readily represented by any objective yielding n .wiw = 1. This 

 would be an immersion lens, the water or balsam angle of which 

 is double the critical angle within water or balsam. But any 

 definite part of the unit can be represented by a dry objective, — 

 for example, the half of the unit by an objective of 60'^ air angle. 



By the value of a any two objectives performing with difi'erent 

 media are directly compared, without needing the introduction of 

 a merely hypothetical angle (as, for instance, the balsam angle of 

 a dry lens) which has no real existence in the performance of the 

 system. 



The numerical value affords at once a clear and exhaustive 

 expression of the relation between any two objectives in respect 

 to all significant functions of the optical performance as far as 

 they depend on aperture ; while by adhering to the angles the 

 judgment would be greatly misled. Comparing two dry lenses 

 of 110° and 140° angular aperture in air, the increase of aperture 

 in the latter seems, according to the angles, equal to 27 per cent. 

 Comparing the true measures (num. ap.) 0*82 and 0-94, the real 

 difference is reduced to not quite 15 jier cent. Again, taking the 

 case of an objective of 1*40 numerical aperture on the homogeneous- 

 immersion system, its balsam angle (138°) looks remote from the 

 ultimate limit of 180°, and there .appears ample range for further 

 increase of aperture and resolving power. But, in fact, the 

 aperture is brought within 7-8 per cent, of the ultimate limit 

 which in every objective is the refractive index of the least refrac- 

 tive medium between the object and the first convex surface of 

 the system. As long as there is not used a substance of much 

 higher refractive power than crown glass as a mounting medium, 

 working medium, and front lens, the utmost increase of the angle 

 of the entering cone of rays would increase very slightly the 

 angle of the image-forming pencil on the other side ; and as from 

 nothing, nothing can come, the advance in performance would 

 remain practically inappreciable. 



From these explanations it will, I hope, bo understood that tho 

 application of the numerical scale in the apertometer is not a mere 

 fancy, but a deliberate attempt to get rid of tho serious defects 

 ap[)ertaining to the method at present in use. 



I add a few remarks relative to the degree of accuracy attained 

 by the apertometer, as I cannot agree with Dr. Woodward in tho 

 objections he makes to the construction of the apparatus on this 

 point. 



Dr. AVoodward is mistaken in supposing errors to bo intro- 

 duced by the reflecting surface of the glass disk. This subject was 

 caivfully exiuuincd before I adopted the arrangement in (juestion. 

 The angle iufliidcd l)y any two symmetrical lines of the division 



