Page:BatemanConformal.djvu/14

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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.