And let it be required to find the Focus where a whole Sphere will collect the Beams proceeding from an Object at the distance d: Here t is equal to 2r, and r equal to ρ. And after due Reduction, the Theorem will stand thus, mpdr - 2ndr + 2nprr / 2nd + 2nr - mpr = f; but if d be Infinite, it is contracted to mpr / 2n - r = 2n - m / 2m - 2n r = f, wherefore a Sphere of Glass collects the Sun-Beams at half the Semi-diameter of the Sphere without it, and a Sphere of Water at a whole Semi-diameter. But if the Ratio of Refraction m to n be as 2 to 1, the Focus falls on the opposite Surface of the Sphere; but if it be of greater Inequality it falls within.
Another Example shall be when a Hemisphere is exposed to parallel Rays, that is, d and ρ being infinite, and t = r, and after due Reduction the Theorem results nn / mm - mn r = f. That is, in Glass it is at 4⁄3r, in Water at 9⁄4r; but if the Hemisphere were Diamant, it would collect the Beams at 14⁄15 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.
APPENDIX.
An Analytical Resolution of certain Equations of the Third, Fifth, Seventh, Ninth Powers, and so on ad Infinitum, in finite Terms, after the manner of Cardan's Rules for Cubicks. By Mr. A. Moivre, Transact. No 309.
LET (n) be any Number, (y) an unknown Quantity, or Root of the Equation, (a) a Quantity altogether known, or what they call Homogeneum Comparationis: And let the Relation of these Quantities to each other be exprest by the Equation.
ny + nn - 1 / 2 × 3 ny3 + nn - 1 / 2 × 3 × nn - 9 / 4 × 5 ny5 + nn - 1 / 2 × 3 × nn - 9 / 4 × 5 × nn - 25 / 6 × 7 ny7, &c. = a.
Its plain from the Nature of this Series, that if n be any odd Number (that is an Integer, it matters not whether Affirmative or Negative) then the Series will Terminate, and the Equation arising will be one of the above defin'd, whose Root is