Some new, powerful, and very simple Bell inequalities are derived. They establish that for two spin-
s particles in the singlet state, the quantum-theoretic Einstein-Podolsky-Rosen correlations for spin measurements along all pairs of directions from a set of
N axes are incompatible with local realism for any set of distinct coplanar axes when
N=3, and for any set of distinct axes whatever when
N=4, with the possible exception of sets restricted by one of the constraints $\stackrel{^}{a}\ifmmode\pm\else\textpm\fi{}\stackrel{^}{b}\ifmmode\pm\else\textpm\fi{}\stackrel{^}{c}\ifmmode\pm\else\textpm\fi{}\stackrel{^}{d}=0$. These results hold for any value of the spin.