Page:Optics.djvu/102

78

which is the focus of the opposite hyperboloid, and conversely, that rays diverging from a point may be refracted by such a lens so as to become parallel.

104.⁠Sir I. Newton has given in the 14th section of the first volume of his Principia, two curious propositions relating to the present subject, which I will insert here, to save the trouble of referring to the book itself.

The first is:

To find the form of the surface of a medium, which will refract rays diverging from a point without it accurately to a point within itself.

Let $A$ (Fig. 101.) be the focus of the incident rays; $B$ that of the refracted; $CD$ the section of the surface; $AD,\ DB$ an incident and refracted ray, $DE$ an infinitely small portion of the curve; $EF,\ EG$ perpendiculars on $AD,\ DB.$

⁠Now ${\frac {DF}{DE}}=\cos EDF=\mathrm {sin~incid.}$

⁠${\frac {DG}{DE}}=\cos EDG=\mathrm {sin~refr.}$

therefore if $m$ be as usual the refracting power, $DF=m.DG;$ but $DF,\ DG$ are the corresponding increments of $AD,\ BD$ so that if we call these $u,\ v,$ we have this differential equation to the curve,

$du=-m.dv,\quad \mathrm {or~~} du+mdv=0$

When $B$ is removed to an infinite distance, the equation becomes $du=mdx$ if $x=AN,$ $DN$ being perpendicular to $AC;$ now we shall see that this equation belongs to an hyperbola, or an ellipse, according as $m$ is greater or less than unity,

$du=mdx$ gives, by integration, $u=mx+n,$

that is, $x^{2}+y^{2}=(mx+n)^{2};$

$\therefore y^{2}=(m^{2}-1)x^{2}+2mx+n^{2}.$

Now the equation to an hyperbola, when the abscissa is measured from the farther focus, is