M. G. Signorelli, G. H. Hoffstaetter
In this report, the general first order particle spin precession in a horizontal storage ring is derived from the Thomas-BMT equation via a perturbative approach of the momentum deviation. Fringe fields are included. Then, by substitution of the equations of motion and orthonormal basis transformations, it is shown how different but equivalent forms of the precession vector appearing in various literature may be obtained. Finally, the utility of certain forms of the spin precession vector is shown by a detailed re-derivation of the spin matching conditions derived by V.I. Ptitsyn for the Electron Storage Ring (ESR) of the Electron-Ion Collider at Brookhaven National Laboratory.