For a Plummer sphere, one can calculate the rotation curve, V_{c}(R) or (R) and from this (R) [ = (R d^{2} /d R + 4^{2})^{½} ]. The figure shows (R) and the two curves + ½ and  ½ . If we now imagine a pattern speed, with frequency _{p} (which is independent of radius), then we can ascertain the radii at which the inner and outer Lindblad radii occur, as well as corotation. Notice in this example there are, in fact, two inner Lindblad resonances.
