# Page:BatemanConformal.djvu/14

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1908.]
83
The conformal transformations of a space of four dimensions.

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

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

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 ${\displaystyle x_{0}y_{0}z_{0}}$, 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

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

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

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

for t; then, since

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

the function F must satisfy the differential equation

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

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

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

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

 ${\displaystyle F(X,\ Y,\ Z,\ T),}$

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

 ${\displaystyle F=0}$

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.