Page:Philosophical Transactions of the Royal Society A - Volume 184.djvu/46

Similarly

$\sin n\phi \int _{0}^{\pi }{\frac {\cos n\phi \;d\phi }{\sqrt {\left(z^{2}+c^{2}-2c\varpi \cos \phi +\varpi ^{2}\right)}}}$

is a solution.

Let $\mathrm {V}$ stand for either of these integrals. Then

${\frac {d^{p+q}\mathrm {V} }{dc^{p}\;dz^{q}}}$

is also a solution of Laplace's equation.

For different values of $p$ and $q$ the solutions are not all independent: as it is easily seen that

$\int _{0}^{\pi }{\frac {\cos \phi \;d\phi }{\sqrt {\left(z^{2}+c^{2}-2c\varpi \cos \phi +\varpi ^{2}\right)}}}$

satisfies a linear partial differential equation of the second order in $c$ and $z$.

But two independent sets of solutions are obtained by giving different values to $p$ in ${\frac {d^{p}\mathrm {V} }{dc^{p}}}$ and $\left({\frac {d}{dc}}\right)^{p}\cdot \left({\frac {d\mathrm {V} }{dz}}\right)\cdot$

The only cases considered in this paper are when $n=0$ or $n=1$. That is the solutions of the forms

$\left({\frac {d}{dc}}\right)^{p}\int _{0}^{\pi }{\frac {d\phi }{\sqrt {\left(z^{2}+c^{2}-2c\varpi \cos \phi +\varpi ^{2}\right)}}}$;

$\left({\frac {d}{dc}}\right)^{p}{\frac {d}{dz}}\int _{0}^{\pi }{\frac {d\phi }{\sqrt {\left(z^{2}+c^{2}-2c\varpi \cos \phi +\varpi ^{2}\right)}}}$;

$\cos \phi \left({\frac {d}{dc}}\right)^{p}\int _{0}^{\pi }{\frac {\cos \phi \;d\phi }{\sqrt {\left(z^{2}+c^{2}-2c\varpi \cos \phi +\varpi ^{2}\right)}}}$;

and

$\cos \phi \left({\frac {d}{dc}}\right)^{p}{\frac {d}{dz}}\int _{0}^{\pi }{\frac {\cos \phi \;d\phi }{\sqrt {\left(z^{2}+c^{2}-2c\varpi \cos \phi +\varpi ^{2}\right)}}}$;

as these cover all the cases to which the functions are applied in this paper.

It is easily seen that

$\varpi \int _{0}^{\pi }{\frac {\cos \phi \;d\phi }{\sqrt {\left(z^{2}+c^{2}-2c\varpi \cos \phi +\varpi ^{2}\right)}}}$

and, consequently,

$\varpi \left({\frac {d}{dc}}\right)^{p}\int _{0}^{\pi }{\frac {\cos \phi \;d\phi }{\sqrt {\left(z^{2}+c^{2}-2c\varpi \cos \phi +\varpi ^{2}\right)}}}$