Page:BatemanConformal.djvu/10

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

${\begin{array}{rcl}l\lambda =&(b-c)(z-a)&=\xi ,\\\\m\mu =&(c-a)(z-b)&=\eta ,\\\\n\nu =&(a-b)(z-c)&=\zeta ,\end{array}}$

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

$P(\xi ,\eta ,\zeta )\equiv P(z)\equiv H\left(\xi ^{b-c}\eta ^{e-a}\zeta ^{a-b}\right)\equiv H(\theta ),$

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

$\xi {\frac {\partial ^{2}P}{\partial \xi ^{2}}}+\eta {\frac {\partial ^{2}P}{\partial \eta ^{2}}}+\zeta {\frac {\partial ^{2}P}{\partial \zeta ^{2}}}+(1-\alpha -\alpha '){\frac {\partial P}{\partial \xi }}+(1-\beta -\beta '){\frac {\partial P}{\partial \eta }}$ $+(1-\gamma -\gamma '){\frac {\partial P}{\partial \zeta }}+\left({\frac {\alpha \alpha '}{\xi }}+{\frac {\beta \beta '}{\eta }}+{\frac {\gamma \gamma '}{\zeta }}\right)P=0$