Page:Optics.djvu/39

15

${\begin{aligned}q\backprime ={\frac {qf}{q+f-qv}}&={\frac {qf}{q+f}}\left(1-{\frac {qv}{q+f}}\right)^{-1}\\&={\frac {qf}{q+f}}\left\{1+{\frac {qv}{q+f}}+\left({\frac {qv}{q+f}}\right)^{2}+....\right.\\&={\frac {qf}{q+f}}+{\frac {q^{2}f.v}{(q+f)^{2}}}+{\frac {q^{3}fv^{2}}{(q+f)^{3}}}+....\end{aligned}}$

16. There is another series a little different from this, but which amounts nearly to the same thing.

If we take the letter $\alpha$ to represent the aberration,

${\begin{aligned}\alpha =q\backprime -q\prime &={\frac {qf}{f+q\cos \theta }}-{\frac {qf}{f+q}}\\&={\frac {qf}{f+q}}\left({\frac {f+q}{f+q\prime \cos \theta }}-1\right)\\&={\frac {qf}{q+f}}.{\frac {q-q\cos \theta }{f+q\cos \theta .}}\\&={\frac {qf}{q+f}}.{\frac {q\operatorname {ver~sin} \theta }{(f+q)\cos \theta +f\operatorname {ver~sin} \theta }}\\&={\frac {q^{2}f.v}{(q+f)^{2}.\cos \theta }}.\left\{1+{\frac {fv}{(q+f)\cos \theta }}\right\}^{-1}\\&={\frac {q^{2}f.v}{(q+f)^{2}.\cos \theta }}\left\{1-{\frac {fv}{(q+f)\cos \theta }}+{\frac {f^{2}v^{2}}{(q+f)^{2}\cos \theta ^{2}}}+\ldots \right.\\&={\frac {q^{2}}{(q+f)^{2}}}f.{\frac {\operatorname {ver~sin} \theta }{\cos \theta }}-{\frac {q^{2}}{(q+f)^{3}}}f^{2}.{\frac {\operatorname {ver~sin} \theta ^{2}}{\cos \theta ^{2}}}+....\end{aligned}}$

Now ${\frac {\operatorname {ver~sin} \theta }{\cos \theta }}={\frac {1-\cos \theta }{\cos \theta }}=\sec \theta -1$; so that $f$ being half the radius, $f{\frac {\operatorname {ver~sin} \theta }{\cos \theta }}$ is the algebraical value of half the excess of the secant over the radius, and if $RT$ be drawn touching the surface at $R$, we shall have the following series in geometrical terms,

$\alpha ={\frac {QE^{2}}{QF^{2}}}.{\frac {AT}{2}}-{\frac {QE^{3}}{QF^{3}}}.{\frac {AT^{2}}{4QE}}+{\frac {QE^{4}}{QF^{4}}}.{\frac {AT^{3}}{8QE^{2}}}-....$