(_12drρ - 3dρt + 9rρt_)/(_4dr + 4dρ - 12rρ - dt + 3rt_) = _f_.
If it could be made of _Diamant_, whose Refraction is as 5 to 2, it
would be
(_(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_)
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.