1884.] 



MICROSCOPICAL JOURNAL. 



169 



Although I have unfortunately not 

 had sufficient opportunity for prop- 

 erly executing an objective of the first 

 mentioned description, and thus prac- 

 tically demonstrating the advantages 

 of the same, I must confess that 

 during the time I have become con- 

 scious of a practical defect of the 

 same, w^hich 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 other- 

 wise be. The objective which I now 

 wish to describe is free from this de- 

 fect. It consists of two lenses only : 

 one of crown and one of flint glass, 

 like the ordinary objective. But the 

 formula for the same is based upon a 

 new principle. 



' 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 theo- 

 retical 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 aper- 

 ture, or in other words, the propor- 

 tion of aperture to focal distance of 

 an objective, is limited by the spheri- 

 cal aberration of the crown-glass lens, 

 because the latter greatly increases 

 with the increase of the angular aper- 

 ture, 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 correspond- 

 ing reduction of the aberrations of the 

 higher order. According to this, w^e 

 can imagine two achromatic object- 

 ives which are equal in focal distance 

 and aperture ; but, although the flint 

 glass lenses of both have the best pos- 

 sible form for correction of the aber- 

 rations of their respective crown-glass 

 lenses, one of the lenses is superior to 



the other in correction of the aberra- 

 tions of higher order, because the 

 spherical aberration of the crown-glass 

 lens is less than that of the other. 

 We now arrive at the question wheth- 

 er the spherical aberration of the 

 crown-glass lens of the present achro- 

 matic objective can be reduced by a 

 mere change of proportion of curva- 

 ture, and, if so, what is the theoreti- 

 cal law after which this proportion 

 has been found? This law, which I 

 have found by careful study, may be 

 expressed as follows : The spherical 

 aberration of a lens for rays of given 

 direction will be a minimum if the 

 proportion of the curvatures of the 

 refracting surfaces is such by which 

 the angle of refraction of the medium 

 ray at the entering surface is equal to 

 that at the emerging ; or, in other 

 words, by which the angle of the 

 perpendicular inclination of the me- 

 dium ray at the entering surface is 

 equal to that of the emerging surface. 

 If the rays entering a lens are parallel 

 or nearly so, as is the case with the 

 telescope, then they will, after having 

 passed through the lens, be changed 

 by refraction to a converging direc- 

 tion toward the focal point of the 

 lens, and to be equal in perpendicular 

 inclination upon their respective sur- 

 faces. The entering or first surface 

 will certainly have to be of corre- 

 spondingly shorter cur\-ature than 

 the emerging or second surface. For 

 a lens of a relative focus and diame- 

 ter, as the crown-glass lens of the 

 present telescope, the radius of the 

 curvature of the inner surface will 

 have to be about twice as long as that 

 of the outer surface to fulfil the con- 

 dition of minimum spherical aberra- 

 tion. But we are familiar enough 

 with the construction of our present 

 objective to admit that just the con- 

 trary is the case ; that is, the curva- 

 ture of the outer surface of the crown- 

 glass lens is by far the longest. If 

 the crown-glass lens is reversed, so 

 that the inner or shorter cui'ved sur- 

 face is brought outside, toward the 

 parallel rays of the object, then the 



