9
Since RE, (Fig. 5.) bisects the angle QRq, we have
QRRq=QEEq.
And in the extreme case
QAAq=QEEq;
that is, if we call AQ, ∆; Aq, ∆′; AE, r as before,
∆∆′=∆−rr−∆′;
so that in fact ∆, r, and ∆′ are in harmonic progression, and we have
1∆+1∆′=2r, or ∆′=∆r2∆−r,
and if Q be supposed infinitely distant, or 1∆=0,
then 1∆′=2r, or ∆′=r2,
which agrees with the preceding formula.
If, as before, we put f for r2, we shall have
∆′=∆f∆−f, and ∆′−f=f2∆−f,
that is, Fq=FE2FQ,
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