We study a general elliptical potential of the form \psi ( x ^ { 2 } + y ^ { 2 } / q ^ { 2 } ) ~ { } ( 0 < q \leq 1 ) plus an additional shear ( with an arbitrary direction ) as models for the observed quadruple lenses .
It is shown that a minimum additional shear is needed even just to reproduce the observed positions alone .
We also obtain the dependence of the axial ratio , q , on the orientation of the major axis of potential .
A general relation also exists between the shear , the position angle and axial ratio of the lensing galaxy .
The relation shows a generic degeneracy in modelling quadruple lenses .
In particular , it shows that only the ratio of the ellipticity , \epsilon \equiv ( 1 - q ^ { 2 } ) / ( 1 + q ^ { 2 } ) , to the magnitude of shear , \gamma can be determined .
All these results are valid regardless of the radial profile of the potential .
Our formalism applies when the galaxy position is observed , which is the case for seven of the eight known quadruple lenses .
Application to these seven cases reveals two quadruple lenses CLASS 1608+656 and HST 12531–2914 , requiring highly significant shear with magnitude \approx 0.2 .
For HST 12531–2914 , there must be a misalignment between the major axis of light and the major axis of potential ( mass ) .
We conclude that detailed modelling of quadruple lenses can yield valuable quantitative information about the shape of lensing galaxies and their dark matter halos .