ZOOLOGY AND BOTANY, MICROSCOPY, ETC. 707 



respective crown glass lens, one of the lenses is superior to the other 

 in the correction of the aberrations 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 whether the spherical aberration 

 of the crown glass lens of the present achromatic objective can be 

 reduced by a mere change of proportion of curvature, and if so, what 

 is the theoretical law after which this proportion must be 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 direc- 

 tion 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 interior surface is equal to that at the emerging ; 

 or, in other words, by which the angle of the perpendicular inclina- 

 tion of the medium ray at the entering siu-face is equal to that of the 

 emerging surface. If the rays entering a lens are parallel or nearly 

 60, as is the case with the telescope, then they will, after having 

 passed through the lens, be changed by refraction to a converging 

 direction toward the focal point of the lens, and to be equal in per- 

 pendicular inclination upon their respective surfaces. The entering 

 or first surface will certainly have to be of correspondingly shorter 

 curvature than the emerging or second surface. For a lens of a 

 relative focus and diameter, 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 

 condition of minimum spherical aberration. But we are familiar 

 enough with the construction of our present objective to admit that 

 just the contrary is the case, that is, the curvature 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 curved surface is brought 

 outside, toward the parallel rays of the object, then the form of the 

 lens would much nearer fulfil the conditions of minimum spherical 

 aberration. But then, of course, the flint glass lens will no longer 

 have the proper form as a correcting lens ; it would now over-correct 

 the spherical aberration of the crown glass lens, and therefore a more 

 flat long curved form of the same would be required. If the exact form 

 or curvature of minimum aberration of the crown glass lens, as well 

 as that of the correcting flint glass lens, as found by calculation, is 

 compared with the present objective, it v/ill bo found that the 

 aberrations of higher order in the new objective are reduced to 

 about one-third of the old one, and a corres^jonding gain in the 

 definition and reduction of colour, or otherwise an extension of tho 

 limit of aperture must be tho result. Let me right here mention 

 another idea as a further step for improvement of the objective in 

 the same direction as described, that is, a further reduction of tho 

 aberrations of higher order. 



" I have in my foregoing description given the law after which 

 a lens of minimum spherical aberration for rays of a given direction 

 has to be constructed, and I will here complete this law by adding 

 that : Tbo absolute minimum of spherical aberration of a lens is 



