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 parallel; that if the luminous point be moved towards the mirror, the focus $q$ will come forwards to meet it, and at the centre $E$, they will coincide, (Fig. 6.) as the formula will easily show, and as one might naturally expect, for the rays in that case being all normal to the surface, would be reflected back upon themselves. When the light is brought between the centre and the surface, or between $E$ and $A$, (Fig. 7.) $Q$ and $q$ in a manner change places, as one might expect from observing that in the formula

$1⁄∆+1⁄∆′=2⁄r$ and $∆$ are quite similarly involved, and therefore may be commuted without altering the equation. When the light is at the middle of the line $∆′$, namely, in the principal focus $EA$, (Fig. 8.) the formula shows that we must have $F$, that is, $1⁄∆′=0$ infinite, which answers to what we found before, namely, that when the incident rays are parallel, or $∆′$ is at an infinite distance, the reflected rays meet in $Q$.

When $F$ is brought between $Q$ and $F$, (Fig. 9.), or $A$ is made less than $∆$, the formula $r⁄2$, or $1⁄∆+1⁄∆′=2⁄r$, shows that $1⁄∆′=2⁄r−1⁄∆$, and consequently $1⁄∆′$, must be negative, that is, $∆′$ goes to the other side of the reflector, and the reflected rays instead of converging, diverge.

When $q$ comes to $Q$, $A$ meets it there.

As a converse to the last case but one, we may take that of rays converging to a point $q$ behind the reflector, and reflected to a focus $Q$ in front, (Fig. 10.) To accommodate the formula to this case, we must make $q$ negative, and we have then

which shows $∆$ to be essentially positive.

The student will find no difficulty in examining particular cases;