# Page:BatemanConformal.djvu/14

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

 $\left(\frac{\partial V}{\partial x}\right)^{2}+\left(\frac{\partial V}{\partial y}\right)^{2}+\left(\frac{\partial V}{\partial z}\right)^{2}=\mu^{2},$

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.[2]

Since V is proportional to the time this differential equation may be replaced by

 $\left(\frac{\partial t}{\partial x}\right)^{2}+\left(\frac{\partial t}{\partial y}\right)^{2}+\left(\frac{\partial t}{\partial z}\right)^{2}=\frac{1}{C^{2}},$

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

 $F(x,\ y,\ z,\ t)=0$

for t; then, since

 $\frac{\partial F}{\partial x}+\frac{\partial F}{\partial t}\frac{\partial t}{\partial x}=0$,

the function F must satisfy the differential equation

 $\left(\frac{\partial F}{\partial x}\right)^{2}+\left(\frac{\partial F}{\partial y}\right)^{2}+\left(\frac{\partial F}{\partial z}\right)^{2}=\frac{1}{C^{2}}\left(\frac{\partial F}{\partial t}\right)^{2}.$

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

 $\begin{array}{clc} X=X(x,\ y,\ z,\ t), & & Z=Z(x,\ y,\ z,\ t),\\ \\Y=Y(x,\ y,\ z,\ t), & & T=T(x,\ y,\ z,\ t),\end{array}$

be the formulæ giving a transformation which enables us to pass from one solution of the above equation to another; then

 $F(X,\ Y,\ Z,\ T),$

when expressed in terms of x, y, z, t, is a second solution of the equation, and if the equation

 $F=0$

be solved for t, the surfaces t = const, will form a system of parallel wave

1. See Herman's Optics, p. 253.
2. Cf. Schwarzschild's Untersuchungen zur Geometrischen Optik, Göttingen Abhandlungen (2), 4.