Boundary pseudomomentum of a sloping Eady layer ($\mathcal{P}=\int \left[\frac{{\psi }_z^{\prime 2}(D)}{2f_0{A}_D}-\frac{{\psi }_z^{\prime 2}(0)}{2f_0{A}_0}\right]dxdy$)

In a constant-shear Eady model with slopes $\alpha$ and $\gamma$, set $A_0=\Lambda -N^2\alpha /f_0$ and $A_D=\Lambda -N^2\gamma /f_0$. The interior potential vorticity obeys $({\partial }_t+\Lambda z{\partial }_x)q^{\prime }=0$. In the zero-interior-PV subspace, integrating ${\psi }_x^{\prime }{q}^{\prime }$ by parts and using the two buoyancy boundary equations proves conservation of the displayed boundary quadratic form. Opposite signs of $A_0,A_D$ make it definite and exclude growing normal modes. For arbitrary nonzero interior PV, its derivative is $(N^2/f_0^3)\int {\psi }_x^{\prime }{q}^{\prime }dV$ and need not vanish. The restriction to zero interior PV is automatic for a growing smooth mode with complex phase speed.