108 Sig. Santini on Achromatic Telescopes. 



readily ascertained, since /3 =j—^-> and the last lenses being in 



contact, c will be = — /3, and since r = - q, y will = - 6. The 

 figure of the lenses, such that the longitudinal spherical aberra- 

 tion may be destroyed, remains to be determined. For this 

 purpose, let X, X', x", denote arbitrary numbers, on which their 

 figure depends \ and, for brevity (as in my Teoria degli Stro- 



menti Ottici, vol. i. No. 104), let ^ = Q , m ^™ ~ l \ ox 



n . ^ 8 (m — 1)8 (m +2) 



— 4 ( m "" - 1 ? 4 ~f m - 2 ms m (2 m + 1) 



V ~ 4 m ~ i ,P ~~ 2 ( m + 2 ) (™-i)' ' ~"2(w + 2) (m-1)' 

 ; /x',v', f ', <7',t', denoting similar functions 



2 (m + 2) (m-1) 

 when the index m relative to the crown glass changes into the 

 index m f relative to the flint. The proper reductions being 

 then made, it will be found that the equation given in 

 No. 108 of the above cited work, in order that the longitudinal 

 spherical aberration in a system of three lenses may be de- 

 stroyed, is brought, in the present case, to the following : — 



b* b 3 

 v* * + m O *$r t m V9 + — O v-V *') = (a), 



in which are the three arbitrary quantities, X, x', X"; two of 

 them, however, being determined at will, the third follows; 

 when, from the well known principles of optics, the rays for 

 each of the surfaces of the lenses will be obtained. 



NUMERICAL EXAMPLE. 



To construct an achromatic object glass, from the foregoing 

 principles, with crown and flint glass of Frauenhofer's ma- 

 nufacture, of which the following are the indices : — 



For the crown mean rays m = 1. 5300001 , 



„ red rays m — dm — 1.521000] 



For the flint mean rays m f = 1.634494 1 , , _ A _ „„■ ' 



j i j r i ^i^^a^ \dm ' = 0.017787 



„ red rays m'-dm' = 1.616707 J 



From which we get £ = 1.650853. Having assumed d = f ; 



b will be = - $, from whence q = 0.072317, r = - 0.072317, 



jS = - c = + 0.0594247, y = - b = 0.3333333. With respect 



to the figure of the lenses, two of the three quantities, X, X', X", 



remaining arbitrary, we shall determine X, X', so that the first 



