Abstract: This paper gives an explicit formula of the S-curvature of (α,β)-metrics F=α\phi(β/α), and proves that if β is Killing 1-form of constant length with respect to α, then S=0. Next, the author studies the S-curvature of Matsumoto-metric F=α²/(α-β) and (α,β)-metrics F=α+εβ+kβ²/α), and obtains that S=0 if and only if β is Killing 1-form of constant length with respect to α. Actually, the author also obtains the condition of above two metrics to be weak Berwaldian. Here \phi(s) is a C^∞ function, α(y)=\sqrt{a_ij(x)yⁱy^j} is Riemannian metric, β(y)=b_i(x)yⁱ is non zero 1-form and ε,k≠ 0 are constants.

Key words: (α,β) -metric, S-curvature, Weak Berwaldian metric