VOL. XLI.] PHILOSOPHICAL TRANSACTIONS. 3^5 



Now, on a clue consideration of this subject, we have found it possible, by 

 proper methods and expedients, to rectify those errors which proceed from the 

 different degrees of refrangibility in different rays, passing from one medium 

 into another ; admitting only this well-known and established principle, on 

 which we ground our reasoning, viz. " That the sines of refraction of rays, 

 differently refrangible, are one to another in a given proportion, when their 

 sines of incidence are equal." Optics, 2d edit. p. 66. And our present design 

 is, to show what advantage this will yield towards improving and perfecting ca- 

 todioptrical telescopes, by making the speculums of glass, instead of metal, in 

 the following manner: let a b c D b f, fig. 3, pi. Q, represent the section of a 

 concavo-convex speculum, whose two surfaces are segments of unequal spheres; 

 call the radius of the sphere, to which the concave side is ground, a ; and the 

 radius of the convex surface, which must be quicksilvered over, e ; let b r be 

 the axis of the speculum, or a line perpendicular to both the surfaces ; where 

 let p be the principal focus, or point where parallel rays of the most refrangible 

 kind are collected, by this speculum ; and a the focus, or point of concourse, 

 of such rays as are least refrangible ; viz. after they have suffered two refrac- 

 tions, at entering into, and passing out of, the concave surface d e p, and also 

 one reflection from the convex surface a b c. If the radius of concavity be 

 greater than the radius of convexity, as we will in the first place suppose, then 

 p will fall nearer the vertex of the speculum than the point a ; and the interval 

 a p will be the greatest aberration, or error, occasioned by the separation, or 

 unequal refraction, of the greatest and least refrangible rays, after their emer- 

 gence from the concave surface fed. Call the common sine of incidence, n ; 

 the sine of refraction of the least refrangible rays, out of a dense medium into a 

 rarer, m ; and of the most refrangible, n* ; then, according to the known and 

 received laws of refraction and reflection, the focal distance of the most refran- 

 gible rays, from the vertex of the speculum, neglecting its thickness, as of little 

 or no moment in the present case, will be found ^ r = p b. And 



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the quantity of the greatest aberration, occasioned by the difl^erent refrangibility 

 of the most and least refrangible rays, p a, will be to the focal distance just 

 mentioned, ? b, as (« — e) x(/* — m) to (a — e) m-{-en ; which quantity, or error, 

 thus obtained, to abbreviate the calculation, call i ; and now let it be required 

 to form a lens, if possible, which, placed at some given point in the axis, be- 

 tween the focus of the most refrangible rays p, and the vertex of the speculum, 

 as H, shall refract not only the rays of the most refrangible kind tending to the 

 point p, but also the rays of the least refrangible kind tending to a, in such a 

 manner, that both sorts shall concur, after such refraction, in some other point 



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