Page:BatemanConformal.djvu/8

a solution of

${\frac {\partial ^{2}U}{\partial x^{2}}}+{\frac {\partial ^{2}U}{\partial y^{2}}}+{\frac {\partial ^{2}U}{\partial z^{2}}}={\frac {1}{c^{2}}}{\frac {\partial ^{2}U}{\partial t^{2}}},$

the function

${\frac {1}{z-ct}}f\left[{\frac {x}{z-ct}},\ {\frac {y}{z-ct}},\ {\frac {r^{2}-1}{2\left(z-ct\right)}},\ {\frac {r^{2}-1}{2c\left(z-ct\right)}}\right]$

is also a solution.

In following up the connection between different solutions, it is convenient to use polar coordinates. Putting

$x=r\ \cos \ \theta \ \cos \ \phi ,\ y=r\ \cos \theta \ \sin \ \phi ,\ z=r\ \cos \theta \ \sin \ \psi ,$ $w=ict=r\ \sin \ \theta \ \sin \ \psi ;$ $X=R\ \cos \ \Theta \ \cos \ \Phi ,\ Y=R\ \cos \Theta \ \sin \ \Phi ,\ Z=R\ \cos \Theta \ \sin \ \Psi ,$ $W=icT=R\ \sin \ \Theta \ \sin \ \Psi ;$

we obtain the relations

$r^{2}=-e^{-2i\Psi },\ R^{2}=-e^{-2i\psi },\ \sin \Theta =cosec\ \theta ,\ \Psi =\phi .$

There is a similar transformation for Laplace's equation.^[1]

If

$X={\frac {r^{2}-a^{2}}{2(x-iy)}},\ Y={\frac {r^{2}-a^{2}}{2i(x-iy)}},\ Z={\frac {az}{x-iy}},$

a solution f(x, y, z) corresponds to a second solution

${\frac {1}{\sqrt {x+iy}}}f\left({\frac {r^{2}-a^{2}}{2(x-iy)}},\ {\frac {r^{2}+a^{2}}{2i(x-iy)}},\ {\frac {az}{x-iy}}\right).$

Putting

${\begin{array}{ccc}x=r\ \sin \theta \ \cos \ \phi ,&y=r\ \sin \theta \ \cos \ \phi ,&z=r\ \cos \ \theta ,\\\\X=R\ \sin \Theta \ \cos \ \Phi ,&Y=R\ \sin \Theta \ \cos \ \Phi ,&Z=R\ \cos \ \Theta ,\end{array}}$

the formulæ of transformation become

$R=ia\ e^{i\phi },\ r=ia\ e^{i\phi },\ \sin \ \Theta =cosec\ \theta .$

The transition from one solution of Laplace's equation to another is now easily effected.

The effects of combining the different transformations belonging to a group of conformal transformations is most easily studied by interpreting

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1. This transformation was given by the author in a Smith's Prize Essay of 1905; it was deduced from a result given by Brill, Messenger of Mathematics (1891), pp. 135-137. If $V=f(x_{1},\dots ,x,\ t)$ is a solution of the differential equation

   ${\frac {\partial ^{2}V}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}V}{\partial x_{2}^{2}}}+\dots +{\frac {\partial ^{2}V}{\partial x_{n}^{2}}}={\frac {1}{a}}{\frac {\partial V}{\partial t}},$

   another solution is given by

   $t^{-{\frac {1}{2}}n}e^{-\left(x_{1}^{2}+x_{2}^{2}+\dots +x_{n}^{2}\right)/4at}\left({\frac {x_{1}}{t}},\ {\frac {x_{1}}{t}},\dots ,\ {\frac {x_{n}}{t}},\ -{\frac {1}{t}}\right).$