706 SUMMARY OP CURRENT RESEARCHES RELATING TO 



which is, the increased number of lenses. I am now of the opinion 

 that any improvement of objectives which requires additional lenses 

 will always be objectionable, however valuable the improvement may 

 otherwise be. 



" The objective which I now wish to describe is free from this 

 defect. It consists of two lenses only, one of crown and one of flint 

 glass, like the ordinary objective. But the formula is based upon a 

 new principle. In my description of the quadruple objective I have 

 spoken of the so-called aberrations of higher order. Let me briefly 

 review this for the better understanding of the following description. 



" We know that the flint glass lens of an objective acts merely as 

 a corrector of both the spherical and chromatic aberrations of the 

 crown glass lens ; but, owing to this double action, the said correc- 

 tion is, even in its best possible form, imperfect in so far as, when 

 the part or zone lying about midway between the centre and the 

 periphery is just right in correction, then the central part leaves a 

 small remnant uncorrected, while the peripheric zone is already over- 

 corrected. These unremovable remnants or so-called aberrations of 

 higher order are the only cause of those imperfections of the achro- 

 matic objective which are dependent on the figure or curvature of 

 the lens, and therefore the best formula for an objective will be that 

 by which these aberrations are mostly reduced. Since the discovery 

 of achromatism nothing has been spared to find by the aid of mathe- 

 matics the best possible form for the flint glass lens for the correction 

 of the aberrations of the crown glass lens ; but for the finding of the 

 proper form, or to better the proportion of curvatures of the crown 

 glass lens itself, there never was a special rule adopted nor theoretical 

 law found after which to obtain the most favorable result. But the 

 calculations were based upon the principle that for any positive 

 crown glass lens a negative flint glass lens can be found, combined 

 with which it will form an achromatic objective in the common sense, 

 and according to this principle no special pains were taken to find the 

 proper form of the crown glass lens. 



" My object in this paper is to show that for the best possible 

 construction of an achromatic objective the proper figure or proportion 

 of curvatures of the crown glass lens is an important factor, submitted 

 to a positive theoretical law, and that, as a consequence of the neglect 

 of this law, the present objective is far from having the best possible 

 form. The angular aperture, or, in other words, the proportion of 

 aperture to focal distance of an objective, is limited by the spherical 

 aberration of the crown glass lens, because the latter greatly increases 

 with the increase of the angular aperture, and consequently the 

 aberrations of the higher order are increased. But this limit can be 

 extended, if the spherical aberration of the crown glass lens can be, 

 without change of focal length and diameter, reduced by a mere 

 change of curvature, because this reduction involves a corresponding 

 reduction of the aberrations of higher order. According to this we 

 can imagine two achromatic objectives which are equal in focal 

 distance and aperture, but although the flint glass lens of both have 

 the best possible form for correction of the aberrations of their 



