Sig. Santini on Achromatic Telescopes. 109 



two lenses shall be isosceles, which will be done by means of the 

 following equations (No. 106, vol. i.) : — 



From these, in the present case, we shall obtain X= 1.G000676, 

 X'=2.233782 • from whence equution(a) will give x"= 2.9 17899. 

 According to these numbers, the radii of the lenses will come 

 out as follows : — 



1st lens, R = R l = 1.06. 



2d lens, R" = R m = 0.0766555 ; double convex. 



3d lens, of flint glass, double concave: R iv = -0.0800292, 

 R v = — 0.1075404. 



The aperture should be determined so that that of the cor- 

 recting lenses does not exceed the half of their focal length. 

 The aperture of the first lens will thus be about = 0.108 : in 

 round numbers = 0.1, which exceeds remarkably what is 

 adopted in practice for the larger object-glasses. 



Let us now see, first, how in this system the chromatic aber- 

 ration is destroyed for the rays nearest to the axis ; wherefore 

 let us assign for the first lens 0.002 ; for the second, 0.002 ; 

 for the third, 0.001 ; and let k, h\ *», fc iU , k iv , /c v represent the 

 distances at which the rays nearest the axis unite after refrac- 

 tion at the first, second . . . sixth surface. From the prin- 

 ciples of dioptrics we shall find, 



For the mean rays. For the red rays. 



k = 3.060000 k™ = 0.0589577 k = 3.094550 A 5ii = 0.0603419 



A 1 = 0.999673 A iv = 0.1809460 A 1 = 1.016946 k™ = 0.1823450 



A 11 = 0.1542802 A* = 3.14150 A» = 0.1575919 ft* = 0.314422 



So the remaining chromatic aberration will be = 0.000272. 



Before proceeding to calculate the spherical aberration, it is 

 as well to destroy this remaining aberration by a Small variation 

 S d given to the distance d. Calling £ k v the correspondent 



variation of Ar v , there will be found £ k r = — r n ,x - 2 $ d ; 

 wherefore we shall have, 



For the mean rays, A* = 0.314150 — 0.889959 Id 

 For the red rays, h? = 0.314422 — 0.805748 I d 



Making these two values of k y equal to each other, there will 



