the Term _prρ_ vanishes, as being finite, which is no part of the other infinite Terms; and dividing the Remainder by the infinite Part _d_, the Theorem will stand thus _pρr_/(_r + ρ_) = _f_, or in Glass, _2rρ_/(_r + ρ_) = _f_. In case the _Lens_ were _Plano-Convex_ exposed to diverging Beams, instead of _pdρr_/(_dr + dρ - prρ_), _r_ being infinite, it will be _pdρ_/(_d - pρ_) = _f_, or _2dρ_/(_d - 2ρ_) if the _Lens_ be Glass. If the _Lens_ be Double-Convex, and _r_ be equal to ρ, as being formed of Segments of equal Spheres, then will (_pdρr_)/(_dr + dρ - prρ_) be reduced to (_pdr_/(_2d - pr_))_f_; and in case _d_ be infinite, then it will yet be farther contracted to ½_pr_, and _p_ being = _n_/(_m - n_), the focal distance in Glass will be = _r_, in Water 1½_r_, but in Diamant ⅓_r_. I am sensible that these Examples are too much for the compleat Analyst, though I fear too little for the less Skilful; it being very hard, if possible, in such Matters, so to write, as to give satisfaction to both; or to please the one, and instruct the other. But this may suffice to shew the extent of our Theorem, and how easie a Reduction adapts any one case to all the rest. Nor is this only useful to discover the _Focus_ from the other proposed _data_, but from the _Focus_ given, we may thereby determine the distance of the Object; or from the _Focus_ and Distance given, we may find of what Sphere it is requisite to take another Segment, to make any given Segment of another Sphere cast the Beams from the distance _d_ to the _Focus_ _f_. As likewise from the _Lens_, _Focus_, and Distance given, to find the _Ratio_ of Refraction, or of _m_ to _n_, requisite to answer those _Data_. All which it is obvious, are fully determined from the Equation we have hitherto used, _viz._ _pdρr_ = _drf + dρf - prρf_, for to find _d_ the Theorem is (_prρf_)/(_rf + ρf - pρr_) = _d_, the distance of the Object. For ρ the Rule is _drf_/(_pdr + df + prf_) = ρ. But for _p_ will be (_drf + dρf_)/(_dρr + fρr_) = _p_, which latter determines the _Ratio_ of Refraction, _m_ being to _n_, as 1 + _p_ to _p_.