the vertical betatron oscillations are considered in Sec- tion IV. While these corrections are different for the muons and protons, they are rather small ( ∼ 1 ppm). The real value of the length of the straight section, L , can slightly differ from L$_{0}$ . In the general case, The problem of taking into account corrections to the g -2 frequency is very important. One of the main prob- lems is an influence of the radial and vertical betatron os- cillations on the average vertical magnetic field. We can consider the case when the velocity of unperturbed mo- tion, v$_{0}$ , coincides with the absolute value of the velocity of perturbed motion. For the latter motion, the average longitudinal component of the velocity is approximately equal to The difference between the real and nominal values of the average angular frequency of spin rotation is given by It is important that Eq. (11) does not depend explic- itly on B . The first term in the r.h.s. of this equation disappears if we define L$_{0}$ = L . In this case, p$_{0}$ is the vertex of a parabola in the momentum space. To find p$_{0}$ and adjust the ring lattice, one can make measurements with proton beams. Three measurements with different values of p are sufficient. The average proton momentum can be kept with radio frequency (RF) cavities put into straight sections of the ring. The longitudinal electric field in the cavities does not influence the spin dynamics. It can be shown that the average magnetic field for the perturbed motion, B$_{p}$ , slightly differs from that for the unperturbed motion, B$_{u}$ : where λ is given by Eq. (12). Approximately, When v$_{0}$$_{ρ}$/v$_{0}$ ∼ v$_{0}$$_{z}$/v$_{0}$ ∼ 0 . 001 , λ ∼ 0 . 01, the correc- tion to the average vertical magnetic field for the betatron oscillations is rather small and may be even negligible. A noncontinuous vertical magnetic field leads to a lon- gitudinal magnetic field on the edges of the magnets. Possibly, the latter field is a reason of the main correction to the g -2 frequency. It was asserted in Ref. [8] that this field causes “the need to know ∫ B · d l for the muons to a precision of 10 ppb”. However, we should take into account that the longitudinal magnetic field cannot be neglected only on small segments of the beam trajectory near edges of magnets. As a result, the above estimate of precision should be decreased by several orders of mag- nitude. The correction for the longitudinal magnetic field can be carefully examined. As curl B = 0 and B$_{φ}$ = ( z/ρ )( ∂B$_{z}$/∂φ ), the longitudinal magnetic field acting on a particle oscillates. When the vertical velocity oscilla- tion (pitch) is given by v$_{z}$/v$_{0}$ = ψ$_{0}$ cos ( ω$_{v}$t + δ ), Condition (3) leading to Eq. (9) should not be ex- actly satisfied. It can be shown that the relation α = 1 leads to the betatron resonance ν$_{x}$ = 1 which results in zeroth frequency of horizontal coherent betatron oscilla- tion (CBO) of the beam as a whole and a loss of the beam [6]. Therefore, the total length of the straight sec- tions should slightly differ from L$_{0}$ so that the CBO tune would be small but nonzero: Typically, in a weak focusing ring α > 1. Eq. (10) re- sults in L < L$_{0}$ . We expect that the CBO tune about 0 . 01 is sufficient to keep the beam. In this case, the appropri- ate choice of the total length of straight sections λ ∼ 0 . 01 reduces the dependence of the spin rotation frequency on the beam momentum by two orders of magnitude. As a result, the use of proton beams for measuring the average magnetic field becomes quite possible. Experimental details depend on the beam momentum. If it is higher than in the completed experiment (see Ref. [4]), the muon lifetime in the laboratory frame increases and the RF cavities may be helpful not only for protons but also for muons to keep the spin coherence. Other- wise, the use of low muon momentum ( ∼ 0 . 3 GeV/ c ) and much higher statistics (see Ref. [7]) may even be more preferable. In this case, the RF cavities are unnecessary for muons. To estimate the correction, we can suppose that ( ∂B$_{z}$ ) / ( ∂l ) ≈ B$_{0}$/b . The length of the considered tra- jectory segment at the magnet edge is b . Calculations where l is the trajectory length and ω ( b ) c = v$_{0}$/R$_{0}$ is the cyclotron frequency into bending sections. Evidently, ∫ B$_{φ}$dl = z$_{0}$B$_{0}$ sin ( ω$_{v}$t + δ ). B$_{φ}$ = z$_{0}$ ∂B$_{z}$ ∂l sin ( ω$_{v}$t + δ ) , z$_{0}$ = ψ$_{0}$ω ( b ) c R$_{0}$ ω$_{v}$ = ψ$_{0}$R$_{0}$ √$_{n}$ , (16) ν$_{CBO}$ ≡ | 1 − ν$_{x}$ | = ∣ 1 − √$_{1 +}$ λ ∣ ≪ 1 , λ = L − L$_{0}$ L$_{0}$ n. (12) III. AVOIDING A BETATRON RESONANCE B$_{p}$ = ( 1 − λ 1 + λ · v 2 0 ρ + v 2 0 z 4 v 2 0 ) B$_{u}$. (15) B$_{p}$ = 1 − n 1 + λ ( 1 + v 2 0 $_{ρ}$+ v 2 0 z 4 v 2 0 ) B$_{0}$, B$_{u}$ = 1 − n 1 + λ B$_{0}$, (14) v$_{φ}$ = v$_{0}$ ( 1 − < v 2 ρ + v 2 z > 2 v 2 0 ) = v$_{0}$ ( 1 − v 2 0 ρ + v 2 0 z 4 v 2 0 ) . (13) Ω ( a ) − Ω ( a ) 0 Ω ( a ) 0 = n · L − L$_{0}$ L$_{0}$ · p − p$_{0}$ p$_{0}$ − n 2(1 − n ) · ( p − p$_{0}$ ) 2 p 2 0 . (11) α = πR$_{0}$ πR$_{0}$ + L − L$_{0}$ . (10) IV. CORRECTIONS TO THE$_{g}$-2 FREQUENCY 3