Translations:The principle of least time/9/en

Let us take an isotropic medium in which the velocity varies continuously so that there is no interface at which reflection and/or refraction could occur. We now show that in such a case, Fermat’s principle holds in its original formulation (i.e., traveltime is a minimum). Let an arbitrary path between two wavefronts be given (Figure 9b). This arbitrary path is called the test path. Let us first figure out what happens between two closely spaced wavefronts. Because the wavefronts are so close together, we might consider the parts of them within a small region to be two parallel straight lines. The test path AC would be a straight line between the two wavefronts, and the flow line AB would be a straight line orthogonal to both wavefronts. If $\theta$ is the angle between the test path and the flow line, if ds is equal to the length of the flow line, and if $d\sigma$ is the length of the test path, then $d\sigma {\rm {=}}{ds/cos}\theta .$.