Hamilton - Theory of Systems of Rays, Reflection from the Hyperbola

  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.

A First Look Into Hamilton's Theory of Systems of Rays

  Following the argument in Hamilton's Theory of Systems of Rays can be somewhat challenging. It starts out with the reflection of light from a mirror and he gives an analytic representation the law of reflection in equation (A). Hamilton appears to be using biradial coordinates with ρ and ρ' being the distances from the foci or poles R and R' below. Let the unit vector m̂ represent the direction of a line in the plane of the mirror and n̂ the direction normal to it. Since the angle of incidence equals the angle of reflection the magnitudes of the components of the directions, ê and ê', of the rays from the point reflection to the two points R and R' are equal except for the sign the projection onto the mirror. This allows us to derive the vector equivalent of equation (A) which is the projection of these vectors onto the direction of an arbitrary line, l.

One can show that the angle of incidence being equal to that of reflection is equivalent to the path length ℓ=a+b being a minimum at the point of reflection as follows. Here the two radii are a and b and the reference points P and Q.

One proceeds by deriving the condition for the direction ê of the line along the mirror which results from changing the point of reflection by a infinitesimal amount. The first two terms of Taylor's series are used to estimate the changes for the radii a and b.

Next we can find the direction the equation for ê satisfies by doing a simple search for the components of the unknown vector.

So the direction we need is that of â-b̂ corresponding to the unit vector ê=(â-b̂)/| â-b̂|.

The biradial coordinates are not as useful as they might be since we can't vary the two radii independently. We know however that changes in position along the mirror don't change the value of the pathlength ℓ locally so we can try using this as one coordinate. Replacing the direction in the differential dℓ with that perpendicular to it and integrating we get another function of the position R, ϑ=a-b.

Both dℓ and dϑ are exact differentials which means that the result of the integration from one point to another depends just on the endpoints and not the path between. One can derive the condition for an exact differential by varying the path of integration.

Using the expressions for da and db as a functions of x and y above we can now show that dℓ and dϑ satisfy this condition in the plane of the reflection.

The curves of constant ℓ and ϑ can be shown to be ellipses and hyperbolas respectively by substituting the functions of a(x,y) and b(x,y) above into their definitions.

Here's a specific example to help tie everything together.

Hopefully this will help illuminate some of Hamilton's 1824 paper on rays.