It has been known for over a decade that many four-image gravitational lenses exhibit anomalous radio flux ratios .
These anomalies can be explained by adding a clumpy cold dark matter ( CDM ) component to the background galactic potential of the lens .
As an alternative , Evans & Witt ( 7 ) recently suggested that smooth multipole perturbations provide a reasonable alternative to CDM substructure in some but not all cases .
We generalize their method in two ways so as to determine whether multipole models can explain highly anomalous systems .
We carry the multipole expansion to higher order , and also include external tidal shear as a free parameter .
Fitting for the shear proves crucial to finding a physical ( positive-definite density ) model .
For B1422+231 , working to order k _ { max } = 5 ( and including shear ) yields a model that is physical but implausible .
Going to higher order ( k _ { max } \ga 9 ) reduces global departures from ellipticity , but at the cost of introducing small scale wiggles in proximity to the bright images .
These localized undulations are more pronounced in B2045+265 , where k _ { max } \sim 17 multipoles are required to smooth out large scale deviations from elliptical symmetry .
Such modes surely can not be taken at face value ; they must indicate that the models are trying to reproduce some other sort of structure .
Our formalism naturally finds models that fit the data exactly , but we use B0712+472 to show that measurement uncertainties have little effect on our results .
Finally , we consider the system B1933+503 , where two sources are lensed by the same foreground galaxy .
The additional constraints provided by the images of the second source render the multipole model unphysical .
We conclude that external shear must be taken into account to obtain plausible models , and that a purely smooth angular structure for the lens galaxy does not provide a viable alternative to the prevailing CDM clump hypothesis .