Hamilton as an astronomer would have been interested in optics for its use in studying the properties of the mirrors used in telescopes. A parabolic mirror can be used to bring a source on its optical axis to a sharp focus. The mirror is the blue curve at the bottom. The red lines are incoming plane wave surfaces from a very distant source and the relative change in action is constant. The green curves are the outgoing surfaces of constant action. The reference surface is the plane z=1 where ℓ=0 and the steps in action were set equal to 1/4 unit.
The focus appears to be quite good. When the incoming rays are slightly off axis the focus is less precise. Below the incoming rays move downward and to the right. The outgoing rays move upward and also slightly to the right coming to an approximate focus to the right of the previous focal point.
So we have an effective focal plane at z=1. The equation for the mirror is z=A(x²+y²) with A=0.25. The focal length of a parabolic mirror is f=1/(4A).
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