of $S^J_{ll'}$ in the spin-1 case described later.

The linear equations are based on the (usually) very linear response [@ketal1; @daresbury], $\Delta S$, of the complex $S$-matrix to small changes $\Delta V$ in the potential. The expression for this is well known in the uncoupled case and is very simple: $$\Delta S_l = \frac{{\rm i} m }{\hbar^2 k}
\int_0^{\infty} (\psi_l(r))^2 \Delta V(r) {\rm d}r.
\label{second}$$ In Eq.\[second\], the $S$-matrix $S_l$ is written in terms of the asymptotic form of the regular radial wave function as $\psi_l(r) 
\rightarrow I_l(r) - S_l O_l(r)$ where $I_l$ and $O_l$ are incoming and outgoing Coulomb wave functions as before. When inverting $S_l(E_k)$ over a series of energies $E_k$, the energy label $E_k$ is implicit in these equations with index $k$ subsumed with orbital angular momentum $l$ to give an overall channel label. In the case of spin-1/2, the $j$ label is also subsumed in the same way [@candm]. Linear algebraic equations for local variations of $\alpha^{(p)}_n $ follow from the minimisation of $\sigma^2$, [@early; @kalinin; @ketal1].

We now present the generalized linear response relationship which applies to the non-diagonal $S$-matrix for spin-1 elastic scattering. The derivation is given in Section \[derivation\]. For any given set 