only for the case where either $v_{\parallel}^{\prime}(\psi_0)=v_{\perp}^{\prime}(\psi_0)=0$, with $\psi_0=\cos^{-1}({\bf p}\cdot{\bf m}_0)$, or ${\bf m}_0=\pm{\bf p}$. The case ${\bf m}_0=\pm{\bf p}$ is very common experimentally as is discussed below. The stability condition $G\geqslant 0$ may be written $$\begin{aligned}
\label{stab}
&&Q^2v_{\parallel}^2+(Qh+\cos 2\alpha_0)(Qh+\cos^2\alpha_0)+h_{\rm
  p}\left\{Qh(1-3\sin^2\phi_0\sin^2\alpha_0)+\cos 2\alpha_0
(1-2\sin^2\alpha_0\sin^2\phi_0)\right\}-\nonumber\\
&&h_{\rm
  p}^2\sin^2\alpha_0\sin^2\phi_0(1-2\sin^2\phi_0\sin^2\alpha_0)
\geqslant  0,\end{aligned}$$ where $v_{\parallel}=v_{\parallel}(\psi_0)$, $h=v_{\perp}(\psi_0)+h_{\rm ext}$ and $Q=\cos\psi_0$. The condition $F\leqslant 0$ takes the form $$\label{F0}
-2(v_{\parallel}+\gamma h)Q-\gamma(\cos
 2\alpha_0+\cos^2\alpha_0)-\gamma h_{\rm
 p}(1-3\sin^2\phi_0\sin^2\alpha_0)
\leqslant 0.$$

We now discuss several interesting examples, the first of these relating to experiments of Grollier [*et al.*]{} [@18] and others. In these experiments the magnetisation of the polarising magnet, the uniaxial anisotropy axis and the external field are all collinear (along the in-plane $z$-axis in our convention). In this case the equation ${\bm\Gamma}=0$, with ${\bm\Gamma}$ given by Eq. (\[Gamma3\]), shows immediately that possible steady states are given by ${\bf m}_0=\pm{\bf p}(\alpha_0=0,\pi)$, corresponding to the switching magnet moment along the $z$-axis. These are the only solutions when $h_{\rm p}=0$. For $h_{\rm p}\neq 0$ other steady-state solutions may exist but in the parameter regime which has been investigated they are always unstable [@13]. We shall assume this is always the case and concentrate on the solutions ${\bf m}_0=\pm{\bf
  p}$. In the state of parallel 