Abstract: Let A be the mod p Steenrod algebra and S the sphere spectrum localized at p , where p>5 is an arbitrary odd prime. In this paper, some important propositions on the May spectral sequence are first given, and then a new nontrivial family of homotopy elements is detected in the stable homotopy groups of spheres by the May spectral sequence and the Adams spectral sequence. The new one is of degree p(p-1)(pⁿ+sp²+sp+s)-7) and is represented by b_n-1g0γ_s in the E₂^s+4,﹡-term of the Adams spectral sequence, where n 4≥ and 3≤ s<p-2. The main theorem obtained in this paper is an obvious generalization of Theorem I in [1].

Key words: Stable homotopy groups of spheres, Adams spectral sequence, Toda-Smith spectra, May spectral sequence