Page:Optics.djvu/93

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Let $r,r',r''....$ be the successive radii, each having its own proper sign as well as magnitude.

$m,m',m''....$ the indices of refraction at the several surfaces.

$\Delta$ the original focal distance.

$\Delta ',\Delta '',\Delta '''$ those after one, two, three, refractions.

Then, if only we neglect the distances between the surfaces along the axis, we shall have

⁠${\frac {1}{\Delta '}}={\frac {m-1}{mr}}+{\frac {1}{m\Delta }},$

⁠${\frac {1}{\Delta ''}}={\frac {m'-1}{m'r'}}+{\frac {1}{m'\Delta '}}={\frac {m-1}{mm'r}}+{\frac {m'-1}{m'r'}}+{\frac {1}{mm'\Delta }},$

⁠${\frac {1}{\Delta '''}}={\frac {m''-1}{m''r''}}+{\frac {1}{m''\Delta ''}}$

⁠$={\frac {m-1}{mm'm''r}}+{\frac {m'-1}{m'm''r'}}+{\frac {m''-1}{m''r''}}+{\frac {1}{mm'm''\Delta }},$

and so on.

When the surfaces are those of lenses, $m'={\frac {1}{m}},\ m'''={\frac {1}{m''}},$ and the equations are reducible to those we have already seen.