Editing Abbe number (section)

==Derivation==
Starting from the [[Lens#Lensmaker's equation|'''Lensmaker's equation''']] we obtain the [[Lens#Thin lens approximation|'''thin lens''' equation]] by dropping a small term that accounts for lens thickness, <math>\ d\ </math>:<ref>{{Cite book |last=Hecht |first=Eugene |title=Optics |date=2017 |publisher=Pearson |isbn=978-1-292-09693-3 |edition=5 ed/fifth edition, global |location=Boston Columbus Indianapolis New York San Francisco Amsterdam Cape Town Dubai London Madrid Milan Munich}}</ref>
:<math> P = \frac{ 1 }{\ f ~} = (n - 1) \Biggl[ \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } + \frac{\ (n-1)\ d ~}{\ n\ R_1 R_2\ } \Biggr] \approx (n - 1) \left( \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right)\ ,</math>
when <math> d \ll \sqrt{\ R_1 R_2\ } ~.</math>

The change of [[refractive power]] <math>\ P\ </math> between the two wavelengths <math>\ \lambda_\mathsf{short}\ </math> and <math>\ \lambda_\mathsf{long}\ </math> is given by
:<math> \Delta P = P_\mathsf{short} - P_\mathsf{\ \!long} = (n_\mathsf s - n_\mathsf \ell) \left( \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right)\ ,</math>
where <math>\ n_\mathsf s\ </math> and <math>\ n_\mathsf \ell\ </math> are the short and long wavelengths' refractive indexes, respectively, and <math>\ n_\mathsf c\ ,</math> below, is for the center.

The power difference can be expressed relative to the power at the center wavelength (<math>\ \lambda_\mathsf{center}\ </math>)
: <math>\ P_\mathsf c\ =  (n_\mathsf c - 1) \left( \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right)\, ;</math>
by multiplying and dividing by <math>\ n_\mathsf c - 1\ </math> and regrouping, get
:<math> \Delta P = \left( n_\mathsf s - n_\mathsf\ell \right) \left( \frac{\ n_\mathsf c - 1\ }{ n_\mathsf c - 1 } \right) \left( \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right)= \left( \frac{\ \ n_\mathsf s - n_\mathsf\ell\ }{ n_\mathsf c - 1 } \right) P_\mathsf c = \frac{\ P_\mathsf c\ }{ V_\mathsf c } ~.</math>
The relative change is [[inversely proportional]] to <math>\ V_\mathsf c\ :</math>
:<math> \frac{\ \Delta P\ }{ P_\mathsf c } = \frac{ 1 }{\ V_\mathsf c\ } ~.</math>