region which a radiant disturbance, initiated at O, can affect in a given time. Each ray of the disturbance by following its natural path, with the ray-velocity proper to its direction at each instant, can travel to that bounding surface in the time; but if it is constrained to follow some other path, it cannot get so far in the time. Thus any point on the ray-path OQ is the farthest point on that path that a ray starting from O and guided by any constraint, could possibly reach in the time; and the disturbance actually reaches that point by travelling along the ray itself. That is, the path of a ray from P to P′ is that path along which the energy of the disturbance, travelling at each instant with the ray-velocity appropriate to its direction, can pass from P to P′ in the least time. This is the generalization, afforded by the theory of undulations, of Fermat's empirical principle[1], which asserted that a ray of light travels from one point to another along such path as would make its time of transit least.
This principle remains precisely a principle of least time for paths from P up to all points P′ such that the successive wave-fronts between P and P′ belonging to a radiant disturbance maintained at P do not develope any singularity along the course of the ray. But when P′ lies beyond a place of infinite curvature (cuspidal edge) on the wave-front the principle becomes merely one of stationary time: in certain cases it may be even a principle of maximum time. A sufficient illustration is afforded by the simple case, of rays diverging from P, which after any series of refractions finally emerge into an isotropic medium as straight rays at right angles to a wave-front S. Let the ray from P to P′ cross this
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wave-front at Q: then by definition the time for the ray from P to Q is the same as the time for the ray from P to any consecutive point Q′ on this wave-front: in comparing the times for the ray PQP′ and a consecutive ray PQ′P′ we have thus only to
- ↑ Cf. Appendix D.
