1908.]
79
The conformal transformations of a space of four dimensions.
where P is a function of the ratio of any two of the quantities lλ, mμ, nν. The quantity U will then be a solution of the equation
${\frac {\partial ^{2}U}{\partial x^{2}}}+{\frac {\partial ^{2}U}{\partial y^{2}}}+{\frac {\partial ^{2}U}{\partial z^{2}}}={\frac {1}{c^{2}}}{\frac {\partial ^{2}U}{\partial t^{2}}}$.


if the relation
$\alpha +\alpha '+\beta +\beta '+\gamma +\gamma '=1$


is satisfied, for it will then be a homogeneous function of degree 1 in (l, m, n, λ, μ, ν).
Let us put
${\begin{array}{rcl}l\lambda =&(bc)(za)&=\xi ,\\\\m\mu =&(ca)(zb)&=\eta ,\\\\n\nu =&(ab)(zc)&=\zeta ,\end{array}}$


where a, b and c are arbitrary constants. The relation
$l\lambda +m\mu +n\nu =0$


is then satisfied, and P becomes a function of z alone. We may thus write
$P(\xi ,\eta ,\zeta )\equiv P(z)\equiv H\left(\xi ^{bc}\eta ^{ea}\zeta ^{ab}\right)\equiv H(\theta ),$


the particular functional form in terms of ξ, η, ζ being chosen to facilitate the calculations. H is clearly a homogeneous function of degree zero in ξ, η, ζ and therefore in l, m, n, λ, μ, ν.
On differentiating equation (1), we obtain
${\frac {\partial U}{\partial l}}={\frac {a}{l}}U+\lambda {\frac {U}{P}}{\frac {\partial P}{\partial \xi }},$
${\frac {\partial ^{2}U}{\partial l\partial \lambda }}={\frac {aa'}{\xi }}U+{\frac {1\alpha \alpha '}{P}}U{\frac {\partial P}{\partial \xi }}+{\frac {\xi }{P}}U{\frac {\partial ^{2}P}{\partial \xi ^{2}}}.$


The differential equation
${\frac {\partial ^{2}U}{\partial l\partial \lambda }}+{\frac {\partial ^{2}U}{\partial m\partial \mu }}+{\frac {\partial ^{2}U}{\partial n\partial \nu }}=0$


will thus be satisfied, if
$\xi {\frac {\partial ^{2}P}{\partial \xi ^{2}}}+\eta {\frac {\partial ^{2}P}{\partial \eta ^{2}}}+\zeta {\frac {\partial ^{2}P}{\partial \zeta ^{2}}}+(1\alpha \alpha '){\frac {\partial P}{\partial \xi }}+(1\beta \beta '){\frac {\partial P}{\partial \eta }}$
$+(1\gamma \gamma '){\frac {\partial P}{\partial \zeta }}+\left({\frac {\alpha \alpha '}{\xi }}+{\frac {\beta \beta '}{\eta }}+{\frac {\gamma \gamma '}{\zeta }}\right)P=0$

