  We review the spin equations for the muon in the 1.45~T muon storage ring at
Brookhaven National Laboratory, which has subsequently been relocated to
Fermilab. Muons are stored in a uniform 1.45~T magnetic field, and vertical
focusing is provided by four sets of electrostatic quadrupoles placed
symmetrically around the storage ring. The storage ring is operated at the
"magic $\gamma = 29.3$" so that the effect of the motional magnetic field
cancels for muons at the magic momentum. We point out the relative sizes of the
various terms in the spin equations, and show that for experiments that use the
magic $\gamma$ and electric quadrupole focusing to store the muon beam, any
proposed effect that multiplies either the motional magnetic field $\vec \beta
\times \vec E$ or the muon pitching motion $\vec \beta \cdot \vec B$ term, will
be smaller by three or more orders of magnitude, relative to the spin
precession from the storage ring magnetic field. We use a recently proposed
General Relativity correction as an example, to demonstrate the smallness of
any such contribution, and point out that the revised preprint from these
authors still contains a conceptual error, that significantly overestimates the
magnitude of their proposed correction. We have prepared this document in the
hope that future authors will find it useful, should they wish to propose
corrections from some additional term added to the Thomas equation, Eq. 13,
below. Our goal is to clarify how the experiment is done, and how the small
corrections due to the presence of the radial electric field and the vertical
pitching motion of themuons (betatron motion) in the storage ring are taken
into account.
