High-quality velocity maps of galaxies frequently exhibit signatures of non-circular streaming motions. We here apply the software tool, velfit recently proposed by Spekkens & Sellwood, to five representative galaxies from the THINGS sample. We describe the strengths and weaknesses of the tool, and show that it is both more powerful and yields results that are more easily interpreted than the commonly used procedure. We demonstrate that it can estimate the magnitudes of forced non-circular motions over a broad range of bar strengths from a strongly barred galaxy, through cases of mild bar-like distortions to placing bounds on the shapes of halos in galaxies having extended rotation curves. We identify mild oval distortions in the inner parts of two dwarf galaxies, NGC 2976 and NGC 7793, and show that the true strength of the non-axisymmetric gas flow in the strongly barred galaxy NGC 2903 is revealed more clearly in our fit to an optical Halpha map than to the neutral hydrogen data. The method can also yield a direct estimate of the ellipticity of a slowly-rotating potential distortion in the flat part of a rotation curve, and we use our results to place tight bounds on the possible ellipticity of the outer halos of NGC 3198 and NGC 2403.
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