to diverging Beams, the Rule is 2 dr ρ / dr + d ρ + 2 r ρ = f , which is affirmative when 2 r ρ is less than dr - d ρ otherwise negative: But in the Case of converging Beams falling on the same Meniscus , 'twill be 2 dr ρ / dr - d ρ + 2 rp = f , and it will be + f , whilst d ρ - dr is less than 2 r ρ; but if it be greater than 2 r ρ, it will always be found negative or - f . If the Lens be double Concave, the Focus of converging Beams is negative, where it was affirmative in the Case of diverging Beams on a double Convex, viz. 2 dr ρ / dr + d ρ - 2 r ρ = f , which is affirmative only when 2 r ρ exceeds dr + d ρ: But diverging Beams passing a double Concave, have always a negative Focus , viz. 2 dr ρ / dr + d ρ + 2 r ρ = - f .
The Theorems for converging Beams, are principally of use to determine the Focus resulting from any sort of Lens placed in a Telescope, between the Focus of the Object-Glass and the Glass it self; the distance between the said Focus of the Object-Glass, and the interposed Lens being made = - d.
I here suppose my Reader acquainted with the Rules of Analytical Multiplication and Division, as that + multiplied by + makes the Product +, + by - makes -, and - by - makes +; so dividing + by + makes the Quote +, + by - makes -, and - by - makes +; which will be necessary to be understood in the preceding Examples.
In case the Beams are parallel, as coming from an infinite distance, (which is supposed in the Case of Telescopes) then will d be supposed Infinite, and in the Theorem pd ρ r