Elasticity/Torsion of rectangular sections

Torsion of a bar with a rectangular cross section
Given:

A cylinder with a rectangular cross sections $$a\times b$$ under torison.

Show:

Why a function of the form

\varphi = m\left(\frac{x_1^2}{a^2}-1\right)\left(\frac{x_2^2}{b^2}-1\right) $$ cannot be used as a Prandtl stress function for this cross section.

Solution
For compatibility, we need

\nabla^2{\varphi} = C $$ However, for the given $$\varphi$$,

\nabla^2{\varphi} = -\frac{2m}{a^2b^2}\left[a^2+b^2-(x_1^2+x_2^2)\right] $$ which is not constant. Hence, the given function cannot be used as a Prandtl stress function.