terms of the coordinates (x, y, z), and the constants of the standard wave front. Then V satisfies the differential equation
and is, in fact, the characteristic function introduced by Hamilton. If it is expressed as a function of x, y, z, and the coordinates of the initial point , it is the Eikonal according to the nomenclature of Bruns.
Since V is proportional to the time this differential equation may be replaced by
where C is the velocity of radiation at the point (x, y, z).
Now suppose that the surfaces t = const, are obtained by solving an equation
for t; then, since
the function F must satisfy the differential equation
Confining ourselves to the case in which C is constant, we may use the results of § 2 to obtain new solutions of this differential equation.
be the formulæ giving a transformation which enables us to pass from one solution of the above equation to another; then
when expressed in terms of x, y, z, t, is a second solution of the equation, and if the equation
be solved for t, the surfaces t = const, will form a system of parallel wave
- See Herman's Optics, p. 253.
- Cf. Schwarzschild's Untersuchungen zur Geometrischen Optik, Göttingen Abhandlungen (2), 4.