The notions of S-curvature and Ξ-curvature introduced by Shen that is very effective for understanding the other Riemannian and non-Riemannian geometric properties of Finsler metrics. Here, we study the S-curvature and Ξcurvature of the class of cubic and quartic (α, β)-metrics. We prove that a third root (α, β)-metric of vanishing Ξ-curvature reduces to a (−1/3)-Kropina metric or it has vanishing S-curvature. Then, we prove that quartic (α, β)-metric of vanishing Ξ-curvature reduces to a special form of quartic (α, β)-metric or it has vanishing S-curvature.
