In this paper, we generalize the product defined by Hardie and Jansen in [3]. The generalized product gives a pairing [·, ·]WΓ: [ΓX∧W, Z] × [ΓY∧W, Z] → [Γ(X∧Y) ∧W, Z]. WhenΓ=S1,X=Sm−1andY=Sn−1, it is the product of Hardie and Jansen. We study properties of it such as bilinearity, Jacobi identity and some formulas in [4]. Furthermore, we consider the dual product and investigate various properties corresponding under the principle of Eckmann-Hilton duality.