(_(10/3)drρ - 2dρt + (4/3)rρt_)/(_5dr + 5dρ - (10/3)rρ - 3dt + 2rt_) = _f_. And this is the universal Rule for the _Foci_ of double Convex Glasses exposed to diverging Rays. But if the thickness of the _Lens_ be rejected, as not sensible, the Rule will be much shorter, _viz._ _pdrρ_/(_dr + dρ - prt_) = _f_, or in Glass _2drρ_/(_dr + dρ - 2rρ_) = _f_, all the Terms wherein _t_ is found being omitted, as equal to nothing. In this Case, if _d_ be so small, as that _2rρ_ exceed _dr + dρ_, then will it be - _f_, or the _Focus_ will be Negative, which shews that the Beams after both Refractions still proceed diverging. To bring this to the other Cases, as of converging Beams, or of Concave Glasses, the Rule is ever composed of the same Terms, only changing the Signs of + and -; for the distance of the Point of Concourse of converging Beams, from the Point B, or the first Surface of the _Lens_, I call a negative Distance or - _d_; and the Radius of a Concave _Lens_ I call a negative Radius, or - _r_ if it be the first Surface, and - ρ if it be the second Surface. Let then converging Beams fall on a double Convex of Glass, and the Theorem will stand thus - _2drρ_/(- _dr - dρ - 2rt_) = + _f_, which shews that in this Case the _Focus_ is always affirmative. If the _Lens_ were a _Meniscus_ of Glass, exposed to diverging Beams, the Rule is - _2drρ_/(- _dr + dρ + 2rρ_) = _f_, which is affirmative when _2rρ_ is less than _dr - dρ_ otherwise negative: But in the Case of converging Beams falling on the same _Meniscus_, 'twill be + _2drρ_/(+ _dr - dρ + 2rp_) = _f_, and it will be + _f_, whilst _dρ - dr_ is less than _2rρ_; but if it be greater than _2rρ_, 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._ - _2drρ_/(+ _dr + dρ - 2rρ_) = _f_, which is affirmative only when _2rρ_ exceeds _dr + dρ_: But diverging Beams passing a double Concave, have always a negative _Focus_, _viz._ - _2drρ_/(+ _dr + dρ + 2rρ_) = - _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 _pdρr_/(_dr + dρ - prρ_)