Page:Optics.djvu/79

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$\therefore m\Delta (\Delta '-r)=\Delta '(\Delta -r);$

$\therefore \Delta '={\frac {m\Delta r}{{\overline {m-1}}\Delta +r}},\quad \mathrm {or} \quad {\frac {1}{\Delta '}}={\frac {m-1}{mr}}+{\frac {1}{m\Delta }}.$

(Case 1, 4.) $m={\frac {\Delta +r}{\Delta }}\cdot {\frac {\Delta '}{\Delta '+r}};$

$\therefore m\Delta (\Delta '+r)=\Delta '(\Delta +r);$

$\therefore \Delta '={\frac {m\Delta r}{-{\overline {m-1}}\Delta +r}},\quad \mathrm {or} \quad {\frac {1}{\Delta '}}=-{\frac {m-1}{mr}}+{\frac {1}{m\Delta }}.$

77. There is another expression sometimes used, in which the distances are measured from the centre, (Fig. 70.)

Let $EQ=q,\quad Eq=q',$

$m={\frac {QE}{AQ}}\cdot {\frac {Aq}{Eq}}={\frac {q}{r-q}}\cdot {\frac {r-q'}{q'}};$

$\therefore q(r-q')=mq'(r-q);$

$\therefore q'={\frac {qr}{mr-{\overline {m-1q}}}},\quad \mathrm {or} \quad {\frac {1}{q'}}=-{\frac {m-1}{r}}+{\frac {m}{q}}.$

78. It will be observed, that we have taken $m$ to represent the ratio of the sines of incidence and refraction in all cases, whether the passage of the light be into a denser or a rarer medium; if we chuse that $m$ should always represent the ratio of the sines of incidence and refraction out of the rarer into the denser, we must, in Cases 3 and 4, put ${\frac {1}{m}}$ for $m$.

Then ${\frac {\Delta \mp r}{\Delta }}\cdot {\frac {\Delta '}{\Delta '\mp r}}={\frac {1}{m}},$ and ${\frac {1}{\Delta '}}=\mp {\frac {m-1}{r}}+{\frac {m}{\Delta }}.$