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Standing fast sausage modes in flare loops were suggested to account for aconsiderable number of quasi-periodic pulsations (QPPs) in the light curves ofsolar flares. This study continues our investigation into the possibility toinvert the measured periods $P$ and damping times $\tau$ of sausage modes todeduce the transverse Alfv\'en time $R/v_{\rm Ai}$, density contrast $\rho_{\rmi}/\rho_{\rm e}$, and the steepness of the density distribution transverse toflare loops. A generic dispersion relation (DR) governing linear sausage modesis derived for pressureless cylinders where density inhomogeneity of arbitraryform takes place within the cylinder. We show that in general the inversionproblem is under-determined for QPP events where only a single sausage modeexists, be the measurements spatially resolved or unresolved. While $R/v_{\rmAi}$ can be inferred to some extent, the range of possible steepness parametersmay be too broad to be useful. However, for spatially resolved measurementswhere an additional mode is present, it is possible to deduce self-consistently$\rho_{\rm i}/\rho_{\rm e}$, the profile steepness, and the internal Alfv\'enspeed $v_{\rm Ai}$. We show that at least for a recent QPP event that involvesa fundamental kink mode in addition to a sausage one, flare loop parameters arewell constrained, even if the specific form of the transverse densitydistribution remains unknown. We conclude that spatially resolved, multi-modeQPP measurements need to be pursued for inferring flare loop parameters.

Inferring flare loop parameters with measurements of standing sausage modes