Page:BatemanConformal.djvu/14

1908.] 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 $$x_{0}y_{0}z_{0}$$, 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.

Let

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