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if, for a light ray, we set then

which is Snell’s law.]
Let us suppose that the properties of the medium slowly change on the scale in
the form

and so the inherent difficulty of this problem is now evident: we seek yet y
appears in the function c. In the case that c = constant, the light rays are straight lines
e.g.

let us seek a solution of (5.81) which satisfies precisely these conditions i.e.

For the given c, (5.82), and with we see that both and are and
so, from (5.81), we have that a cursory analysis of the problem suggests
that we write the solution in the implicit form

where Thus

and we will assume that is such that A, and all its relevant derivatives, lead to
uniform asymptotic expansions in the domain where the material exists. This ensures,
for example, that we may write

for all X, Y in the domain. Differentiation of (5.84) yields

which can be solved for (given from (5.85)). We seek a solution in the form