Similarly
is a solution.
Let stand for either of these integrals. Then
is also a solution of Laplace's equation.
For different values of and the solutions are not all independent: as it is easily seen that
satisfies a linear partial differential equation of the second order in and .
But two independent sets of solutions are obtained by giving different values to in and
The only cases considered in this paper are when or . That is the solutions of the forms
;
;
;
and
;
as these cover all the cases to which the functions are applied in this paper.
It is easily seen that
and, consequently,