112 Sig, Santini on Achromatic Telescopes. 



Making an equation between the values of /c v , we obtain 

 $d = + 0.014013, Jd'= - 0.0009021 ; and the correcting 

 lenses being previously supposed in contact, the negative result 

 is inadmissible. I do not wish to deduce generally from this, that 

 with no other ratio of dispersion the chromatic and spherical 

 aberrations may not even this way be destroyed ; only it appears 

 that they vary considerably with the distance £ d' ; so that, if 

 even in other cases the thing be possible, the micrometrical ap- 

 paratus should be constructed with the greatest care. 



Not having succeeded, by varying the distances, in destroy- 

 ing the spherical aberration, I have had recourse to small 

 arbitrary variations, d R iv , d R v , given to the rays of the last 

 lens, which I have regarded as positive, although relative to a 

 concave lens, having with greater facility obtained in this 

 hypothesis the general equations relative to the path of a ray 

 of light. In this way I have found the following equations of 

 condition. 



Forthe nearest meanrays,Av = 0.317137 — 10.07444 dR iv — 5.51794 dR* 

 For the nearest red rays, & T = 0.317121— 9.79024 c?R iv — 5.37508 dR» 

 Forthe remote mean rays, £ v = 0.316238 — 12.15546 dR iT — 4.97661 rfRv 



The resolution of which gives d R iv = - 0.0002655; 

 d R v = + 0.0006401 ; k v = 0.316280. From whence the 

 corrected values of R iv , R v will be R iv = 0.0797637; 

 R v = 0.1081805 ; the distance remaining d = 0.663437. 



If now the distance k v of the point in which the rays unite 

 behind the last lens be directly calculated in this system, there 

 will be found — 



For the mean rays nearest the axis, k v = 0.316145 

 For the red rays nearest the axis, k v = 0.316136 

 For the mean remote rays, k 9 = 0.316120 



For the red remote rays, A* = 0.315466 



Remaining aberrations. 

 Chromatic in the mean rays, = 0.000009 

 Spherical in the mean rays, = 0.000025 

 Spherical in the red rays, = 0.000669 



The first two remaining chromatic and spherical aberrations 

 are extremely minute, and without the small errors from the 

 tables to seven places of figures, or without the influence of the 



