11973 aad SUNY a t Stony 3rook, Stony Brook, N. Y. -- ABsmcT In t h i s paper we m i n e the nchain of f a s t rerromuce i n t e r e s t is m t drpedeat rather on tb size of tb resonance. I h . in the AGS a t Brookhavea National Laboratory. INTROIWCTION The major problem which must 1# wereuate fir the aceeXaratlPa of polarized protoll. is tb. W O Z l a Z l t dfEpokrig.tttXl t b t OCCWZS ha th. perturbing f i e l d s , as men by the particles, coll.ra;ln ca~pl~aentrr wdtb f requamy equal to the

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... pin preceesim frequsaegr. T b stroogcllve af tbse resonances occurs when where n Y(g/2-1) i a the spia precsslrioa freqztrmey, k f s a intet~gcar,. P is the periodicity of the accelerator, and v i s the v e r t i c a l betatrcw ~wm. s The standard methodl~2 f o r dealing d.Lh these so call4 k r f f h h e resonances is ta "jmpW them by changbg tb tune abruptly as tc approaches a resonance during n o W acceleration. TO study the depolaritatlon due to such a resonance jump, we w2U co'ilgider the following realist* ⪙: l e t no be a m isolated resanenee, and let vary in a three step pmmss: -to -6, very s l~~l y , adiabatically -6 t o 6 , f a s t j q , d(n-no)/dOa (2) 6 t o -, again adiabatically. It is well h0wn3s4 t h a t the equations of motion f o r the spin of p a r t i c l e during acceleratiao can be written, w i t h the aid of spinors, a s where 8 is the turning angle, E is the width of an isolated resonance and i s the frequency of the resonance. The spin vector is given by where a i is the appropriate Pauli matrix. Equation (3) is formally just the Schrbdinger equation with Rarniltonian H. Work performed under the auspices