terms of the coordinates (*x, y, z*), and the constants of the standard wave front. Then *V* satisfies the differential equation^{[1]}

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.^{[2]}

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