  Let L be a normally generated line bundle on X; we say L satisfies property
N_p (notation after Mark Green) if the matrices in the free resolution of R
(the homogeneous coordinate ring of X) over S (the homogeneous coordinate ring
of the projective space corresponding to the complete linear series |L|) have
linear entries until the p-th stage. In this article we prove the following
result: Let X be an elliptic ruled surface and let L be a product of p+1 base
point free and ample line bundles on X. Then L satisfies property N_p. In
particular we prove that numerical classes of all divisors which satisfies
property N_p form a convex set. (Recall that Num(X) is generated by the class
of a minimal section C_0 and by the class of a fiber f and that C_0 is ample.)
As a corollary of the above result we show that the adjoint bundle K_X+(2p+3)A
satisfies property N_p, if A is an ample line bundle.
