- mn r = f . That is, in Glass it is at 4 ⁄ 3 r , in Water at 9 ⁄ 4 r ; but if the Hemisphere were Diamant, it would collect the Beams at 1 4 ⁄ 15 of the Radius beyond the Center.
Lastly, As to the Effect of turning the two sides of a Lens towards an Object; it is evident, that if the thickness of the Lens be very small, so as that you neglect it, or account t = 0; then in all Cases the Focus of the same Lens, to whatsoever Beams, will be the same, without any difference upon the turning the Lens: But if you are so curious as to consider the thickness, (which is seldom worth accounting for) in the Case of parallel Rays falling on a Plano-Convex of Glass, if the plain side be towards the Object, t does occasion no difference, but the focal distance f = 2r. But when the Convex-side is towards the Object, it is contracted to 2r - ⅔t, so that the Focus is nearer by ⅔t. If the Lens be double Convex, the difference is less; if a Meniscus, greater. If the Convexity on both sides be equal, the focal length is about ⅙t shorter than when t = 0. In a Meniscus the Concave-side towards the Object increases the focal Length, but the Convex towards the Object diminishes it. A General Rule for the difference arising on turning the Lens, where the Focus is Affirmative, is this
2rt - 2ρt / 3r + 3ρ - t, for double Convexes of differing Spheres. But for Menisci
the same difference becomes
2rt + 2ρt / 3r - 3ρ + t; of which I need give no other Demonstration, but that by a due Reduction it will so follow from what is premised, as will the Theorems for all sorts of Problems relating to the Foci of Optick-Glasses.