Page:Optics.djvu/32

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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/q^{[errata 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

${\frac {1}{q'}}={\frac {2}{r}},$^[1] ⁠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.

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1. Taking the general formula in this case, we find
   
   ${\frac {1}{q'}}={\frac {2\cos \theta }{r}};$ that is,$q'={\frac {r.\sec \theta }{2}},$
   
   

   so that, generally speaking, when the incident ray is parallel to the axis of the reflector, the reflected ray bisects the secant of the angle, which, in the more particular case of a ray nearly coincident with the axis, becomes the radius.

Errata

1. Original: r was amended to q: detail