c 18 1 
completed, I interchanged with Mr. Dollond for 
his theorem, he taking mine, and I taking his. 
Our theorems, though fimilar, were not exactly the 
fame; but, by reduction to the fame form, I inferred 
his theorem from mine ; which gave me a farther 
confidence of the exadtnefs of both. 
I have here fent you my theorem, and defire, that 
you will lay it before the Royal Society. 
I am, S I R, 
Your moil obedient 
humble fervant, 
Nevil Mafkelyne. 
Prince Henry, St. Helen’s Road, 
Jan. 16, 1761. 
L ET the form of the lens afiumed, in the invefti- 
gation of the theorem, be a menifeus, the radius 
of whofe convex furface is greater than that of its 
concave furface ; and the center of whofe two fur- 
faces lie on the fame fide of the lens, as the radiant 
point, from which the rays diverge, that fall thereon. 
The ray falling on the extreme part of the lens will, 
after refradtion, diverge from a point before the lens, 
nearer thereto than the geometrical focus of rays di- 
verging from the fame radiant point, and pafiing 
indefinitely near the vertex. 
Let Q_exprefs the diflance of the radiant point, 
before the lens, from its vertex ; R, the radius of 
concavity of the furface, on which the rays firfi: fall ; 
and r, the radius of convexity of the fecond furface ; 
F, the principal focus, or the focus of parallel rays ; 
6 which 
