Sig. Santini on Achromatic Telescopes. 107 



lenses conjointly can be brought nearer or removed from the 

 first ; and the errors arising from the figure will be destroyed 

 by a small motion tending to separate in a small degree the 

 two correcting lenses, without its being necessary, as in the 

 ordinary theory, to retouch the surface of the last lens. 



The simplicity of this construction made me curious to cal- 

 culate numerically the dimensions assigned by theory, so as to 

 verify the simple method which is given for destroying the 

 remaining chromatic and spherical aberrations. Having, in 

 this last particular, obtained results which do not exactly agree 

 with the statements of the illustrious author, I have brought 

 them together here, with the results of my calculation, from 

 which it will more clearly appear, under what circumstances, 

 and with what precautions, recourse must be had to the pro- 

 jected correcting lens. 



Imagine a system of three lenses placed in the same axis, 

 and constructed of crown and flint glass, the indices of mean 

 refraction of which are respectively m, m! ; the first and the 

 second being of crown glass and convex ; the third of flint and 

 concave. Let their focal lengths be p, q, r, and p = 1 ; also, 

 let the distances of the points of union of the rays be indicated 

 respectively by a, a ; 6, /3 ; c, y ; the distance of the first from 

 the second lens = d ; let the second and the third be imagined 

 in contact. As the correcting lens should produce in the rays 

 of mean refrangibility the effect of a plane glass, q + r will 

 be = 0, that is, q = — r. Now, considering that the rays 

 parallel to the axis, a case which takes place in the object- 

 glasses designed for astronomical observations, will be a — oo, 

 a zrp = 1, 6 = — (I — d)\ and assuming, for the sake of 



brevity, t = -t — ♦ —, — 7' the equation, which should exist to 

 1 * a m m' - 1 



destroy the longitudinal aberration of refrangibility, will be 



1 + — (I-?) = ; from whence q = 6* (£ - 1) ; 



r = — b* (£ — 1); whence d and b being assumed arbitrarily, 

 the focal distances of the two correcting lenses will be obtained, 

 from which the rule laid down by Mr. Rogers evidently fol- 

 lows, b, q, r, being obtained, the other distances /S, c, 7, are 



