Page:Optics.djvu/32

 9.It appears then, that the place of the point $$q,$$ or the distance $$Eq$$ depends on the value of $$\theta$$, the angle $$REA$$, and is therefore not the same for different rays $$QR.$$ It diminishes as $$\cos \theta$$ increases, that is, as $$\theta$$ diminishes, or $$QR$$ approaches to coincidence with $$QA$$. It is important to know what its final value is, which is in fact to determine the point of intersection of the reflected rays when the incident rays are nearly coincident with $$QA,$$ the axis of the surface, forming consequently a very small pencil.

If we suppose $$\theta=0$$, we shall have $$\cos \theta=1,$$ and $$\frac{1}{q'} =$$ $1⁄1$$$+\frac{2}{r},$$or $$q'=\frac{qr}{r+2q}.$$

10.If moreover we suppose $$q$$ infinite, which is supposing the rays parallel, we shall have

$\frac{1}{q'} = \frac{2}{r},$ or $q'=\frac{r}{2}.$

This is what is technically termed, the principal focal distance of the reflector, the place of $$q$$ being then $$F,$$ which is called the principal focus, and if we call $$AF, f,$$ we shall have in general, that is, for rays nearly coincident with $$QC,$$

$$\frac{1}{q'} = \frac{1}{q}+\frac{1}{f}.$$

11.These formulæ might easily have been obtained directly by supposing $QR$ equivalent to $QA$, and we will make use of this method to deduce them in another form which is often more convenient. Errata