the actual radiation sent on from all points of the element of the wave-front F situated at O, which is tangential to that wave-surface, will reach Q in the same phase, as follows from the definition of the wave-surface and the reversibility of the radiation: hence the effects due to all parts of this element of wave-front situated at will reinforce each other at Q, while those of any other element of the same order of magnitude will obliterate each other owing to differences of phase: thus it is only the portion of the wave-front around that sends radiation to Q, and the other parts of it may be shut off by screens without altering the effect at Q. It is here tacitly assumed that the medium is homogeneous, so that wave-surfaces of all magnitudes round are similar, and the ray OQ is therefore a straight line. When there is heterogeneity we must take a wave-surface of very small dimensions, corresponding to a very short time of transit, so that OQ is an element of arc of the ray; the next element of arc starting from Q will now be in an infinitesimally different direction; and thus the ray will be a curved line. The path of a ray between two points P and P′ is of course actually explored by placing a source at P and gradually limiting the beam by screens so as not to affect the illumination at P′: so long as the screens do not cross the curve PP′ constructed as above this will not be affected.
It remains to express these kinematical ideas in analytical form. With a view to this object, we must distinguish, when the medium is of crystalline quality, between the wave-velocity and the ray-velocity corresponding to any given direction. Thus in the diagram the plane wave-front F is propagated in the direction normal to itself with the wave-velocity appropriate to that direction: but the radiant energy of that wave travels in the direction OQ with the ray-velocity, which is greater than the former in the ratio of OQ to OT, where OT is perpendicular to the tangent plane at Q. When the form of the wave-surface round is known, the wave-velocities and ray-velocities corresponding to all directions are thereby determined.
Now the wave-surface S marks the outer boundary of the
