Back to Hamilton's Theory of Systems of Rays. How does one explain Snell's law of refraction where the index of refraction n=sini/sinr? Mathematically it's just a description of the relation between the paths of the incident ray and the refracted ray. Is there anything significant about the length of the lines corresponding to the sines? They're the altitudes of the triangles with vertices i, n̂₁, and the origin and that for r, n̂₂ and the origin but the ratio of the areas of these triangles is also equal to the index of refraction.
These triangles are isosceles so we could draw the altitude as perpendicular to the rays instead.
What we want is some sort of physical explanation for the law of refraction. We can find this in Feynman's Lectures on Physics, Vol I where the index of refraction is attributed to a phase change due to secondary waves caused by forced oscillations of the electrons in a plate of the transparent body and not a change in the speed of light in the refracting medium. The derivation assumed that the index of refraction differed by a small amount from that of a vacuum which is 1. The relation for a denser medium is found in Vol II. This explanation is essentially that of the classical dispersion theory introduced by Paul Drude a little over a century ago. A formula for normal dispersion can be found in his Lehrbuch der Optik of 1900.
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