# Page:BatemanConformal.djvu/8

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

a solution of

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

the function

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

is also a solution.

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

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

we obtain the relations

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

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

If

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

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

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

Putting

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

the formulæ of transformation become

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

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

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 ${\displaystyle V=f(x_{1},\dots ,x,\ t)}$ is a solution of the differential equation
 ${\displaystyle {\frac {\partial ^{2}V}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}V}{\partial x_{2}^{2}}}+\dots +{\frac {\partial ^{2}V}{\partial x_{n}^{2}}}={\frac {1}{a}}{\frac {\partial V}{\partial t}},}$

another solution is given by

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