  When Schema R fails this is because the mathematics of the particular limit \((\varepsilon \rightarrow 0)\) is singular. One can ask what, physically, is responsible for this mathematical singularity. In investigating the answer to this question one will often find that the mathematical blow-up reflects a physical impossibility. For instance, if Schema R held when \(T_f\) is the wave theory of light and \(T_c\) is the ray theory (geometrical optics), then one would expect to recover rays in the shortwave limit \(\lambda \rightarrow 0\) of the wave theory. On the ray theory, rays are the carriers of energy. But in certain situations families of rays can focus on surfaces or lines called “caustics.” These are not strange esoteric situations. In fact, rainbows are, to a first approximation, described by the focusing of sunlight on these surfaces following its refraction and reflection through raindrops. However, according to the ray theory, the intensity of the light on these focusing surfaces would be infinite. This is part of the physical reason for the mathematical singularities. See also the discussion of the rainbow by Pincock 2011, and Belot 2005.