Hamilton discusses the reflection of a ray from a hyperbolic mirror in Theory of Systems of Rays, Section 1, item 5. This appears to be what he is talking about. Light from focal point P travels to the intersection point R and is reflected by the hyperbola beyond the elliptical mirror. The tangent and normal directions are reversed for the hyperbola. The path of the light reflecting from the other side of the hyperbola is along the extension of the initial ray from P to R.
One can find more information about ellipses and hyperbolas in Casey, A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections.
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