  We derive the first order phase space dependence of spin-orbit motion of a
particle in an accelerator by expanding the Thomas-BMT equation. Different
forms can be found in the literature and we show how these are related, and
care is taken to include fringe fields. The advantages of using certain forms
is demonstrated by a detailed re-derivation of the spin matching conditions by
V. Ptitsyn for the spin rotators in the Electron Storage Ring (ESR) of the
Electron-Ion Collider (EIC) at Brookhaven National Laboratory.
