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9

Since $RE$ , (Fig. 5.) bisects the angle $QRq$ , we have

$QR / Rq = QE / Eq$ .

And in the extreme case

$QA / Aq = QE / Eq$ ;

that is, if we call $AQ$ , $∆$ ; $Aq$ , $∆'$ ; $AE$ , $r$ as before,

$∆ / ∆' = ∆- r / r -∆'$ ;

so that in fact $∆$ , $r$ , and $∆'$ are in harmonic progression, and we have

$1 / ∆ + 1 / ∆' = 2 / r$ , or $∆'= ∆ r / 2∆- r$ ,

and if $Q$ be supposed infinitely distant, or $1 / ∆ =0$ ,

then $1 / ∆' = 2 / r$ , or $∆'= r / 2$ ,

which agrees with the preceding formula.

If, as before, we put $f$ for $r / 2$ , we shall have

$∆'= ∆ f / ∆- f$ , and $∆'- f = f 2 / ∆- f$ ,
that is, $Fq = FE 2 / FQ$ ,

from whence it appears, that

$Fq : FE :: FE : FQ$ .

12. Upon the whole we may collect that if a small luminous body be placed before a spherical concave mirror, at some distance from it, the distance of the focus will always be something more than half the radius of the surface, which is its accurate value for the light of the Sun, the rays of which are considered as