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cases, and to show that the conjugate foci are convertible, as in reflexion, and that what are incident rays in one point of view, may be considered in another as refracted, and vice versâ.

84. It will be observed that in this, and in all other cases of refraction, the conjugate foci move in the same direction, whereas in reflexion they always come towards, or recede from each other.

The following are corresponding values of $\Delta$ and $\Delta '$

$\Delta =\infty$⁠$r,$⁠0,⁠$-{\frac {r}{m-1}}$, $x$

$\Delta '={\frac {mr}{m-1}}$⁠$r,$⁠0,⁠$\infty$,⁠$-{\frac {mr}{m-1}}.$

Case 2.⁠Here we have ${\frac {1}{\Delta '}}=-{\frac {m-1}{mr}}+{\frac {1}{m\Delta }};$ whence it appears that as long as $\Delta >{\frac {r}{m-1}},$ or $Q$ beyond $f,$ (Fig. 76.) $\Delta '$ is negative, or $q$ on the contrary side of $A$ from $Q.$

When $\Delta ={\frac {r}{m-1}},$ or $Q$ is at $f,$ $\Delta '$ is infinite.

${\displaystyle }$When $\Delta <{\frac {r}{m-1}},$ or $Q$ is between $A$ and $f,$ $\Delta '$ is positive; so that $Q$ and $q$ are on the same side of $A:q$ is at first infinitely distant, and its change of place must be very much quicker than that of $Q$, for while this moves from $f$ to $A$, $q$ comes from an infinite distance to the same point.

When $\Delta$ is negative, or $Q$ within the denser medium, Fig. 79.

${\frac {1}{\Delta '}}=-{\frac {m-1}{mr}}-{\frac {1}{m\Delta }}$

$\Delta '$ is then necessarily negative, as we might expect, the two foci moving together from $A$ in the same direction.

$Aq$ is at first greater than $AQ,$ but the two points coincide in $E,$ and afterwards $Q$ gets beyond $q,$ and, in fact, it moves from $E$ to an infinite distance while $q$ goes from $E$ to $F$.