108 Mr. Arry on the use of Silvered Glass 
These investigations were prepared for the notice of the 
Philosophical Society, when I discovered that the same idea had 
been published in the Philosophical Transactions for 1740, by 
Mr. Caleb Smith. He has given without demonstration some for- 
mulz for the correction of colour, but the correction of spherical 
aberration he leaves, as leading into too intricate investigations ; 
the achromatism of the eye-piece does not seem to have occurred 
to him. He speaks of having made one experiment which gave 
him the greatest hopes of success. As the construction has not 
been remarked by any of the friends to whom I have mentioned 
it, and as the subject is one of considerable interest, I am induced 
to think that a statement of the principles, and an investigation 
of the formule for the construction of telescopes on this plan will 
be not unacceptable to the Philosophical Society. 
(1). In the following articles we shall always consider the 
radii of convex surfaces as positive. We shall denote by x the 
ratio of the sine of incidence to the sine of refraction out of air 
into glass for mean rays, and by 6” the variation of this ratio 
arising from unequal refrangibility. For the alteration in focal 
lengths, &c. arising from chromatic aberration we shall use 6, and 
for those produced by spherical aberration we shall use d. Our 
approximation will be extended to the first power of én, and the 
second power of the apertures; and all our investigations will be 
made on the supposition that the telescope is Gregorian. 
(2). A mirror is formed of a double convex lens silvered on 
one surface: rays are incident from a given point in the axis; to 
find the focus of reflected rays. 
(3). 1%. For first refraction. Let C be the center of the sur- 
face 4D; B the given point; BD an incident ray, DE the refracted 
