Elasticity/Torsion of rectangular sections

Given:

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

Show:

Why a function of the form

${\displaystyle \varphi =m\left({\frac {x_{1}^{2}}{a^{2}}}-1\right)\left({\frac {x_{2}^{2}}{b^{2}}}-1\right)}$

cannot be used as a Prandtl stress function for this cross section.

For compatibility, we need

${\displaystyle \nabla ^{2}{\varphi }=C}$

However, for the given ${\displaystyle \varphi }$,

${\displaystyle \nabla ^{2}{\varphi }=-{\frac {2m}{a^{2}b^{2}}}\left[a^{2}+b^{2}-(x_{1}^{2}+x_{2}^{2})\right]}$

which is not constant. Hence, the given function cannot be used as a Prandtl stress function.