1908.]
77
The conformal transformations of a space of four dimensions.
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
${\frac {1}{zct}}f\left[{\frac {x}{zct}},\ {\frac {y}{zct}},\ {\frac {r^{2}1}{2\left(zct\right)}},\ {\frac {r^{2}1}{2c\left(zct\right)}}\right]$


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


a solution f(x, y, z) corresponds to a second solution
${\frac {1}{\sqrt {x+iy}}}f\left({\frac {r^{2}a^{2}}{2(xiy)}},\ {\frac {r^{2}+a^{2}}{2i(xiy)}},\ {\frac {az}{xiy}}\right).$


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
 ↑ 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. 135137. 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
$t^{{\frac {1}{2}}n}e^{\left(x_{1}^{2}+x_{2}^{2}+\dots +x_{n}^{2}\right)/4at}\left({\frac {x_{1}}{t}},\ {\frac {x_{1}}{t}},\dots ,\ {\frac {x_{n}}{t}},\ {\frac {1}{t}}\right).$

