Page:Optics.djvu/45

 principal co-ordinate axis. Let $AN$ (Fig. 20.) be this axis, $AM$, $MP$ co-ordinates of the curve, $PN$ a normal, $QP$, $Pv$ an incident and a reflected ray. The question is first to determine the equation of the line $Pv$. Since this line passes through the point $P$ whose co-ordinates are $x$, $y$, the equation must be

$Y−y=α(X−x)$ being the tangent of the angle $α$.

Now,

And since $PvN$ is a normal,

The equation is therefore

and we have to put for $tanPvN=−tanvPQ=−tan2NPQ=−tan2PNv$ its value in terms of the co-ordinates given by the equation to the curve, and eliminate $PN$ and $tanPNv=dx⁄dy$ between this, the equation (1), and its derivative.

26. The process is sometimes facilitated by taking for the variable a function of the angle $∴ tan2PNv=2dx⁄dy⁄1−dx^{2}⁄dy^{2}=2dxdy⁄dy^{2}−dx^{2}$, as its tangent which is equal to $Y−y+2dxdy⁄dy^{2}−dx^{2}(X−x)=0$. The quantity we have called α is the tangent of twice this angle, and if we put $dxdy⁄dy^{2}–dx^{2}$ for this angle, the equation to the reflected ray is

. Suppose the curve to be a common parabola, its equation is $x$,