  We study the equations for the quasi-momenta which characterize the
wave-functions in the Bethe ansatz for the XXX-Heisenberg model. We show in a
simple analytical fashion, that the usual ``string hypothesis" incorrectly
predicts the number of real solutions and the number of complex solutions for
$N>21$ in the sector with two spins flipped, confirming the work of Essler et
al. Two complex pair solutions drop out and form two additional real pair
solutions. We also introduce a new set of variables which allows the equations
to be written as a single polynomial equation in one variable. We consider in
some detail the case of three spins flipped.
