Let A be the mod p Steenrod algebra for p an arbitrary odd prime and S the sphere spectrum localized at p. In this paper, some useful propositions about the May spectral sequence are first given, and then, two new nontrivial homotopy elements ${alpha}_1{jmath}{xi}_n;(p{geq}5,n;{geq};3);and;{gamma}_s{alpha}_1{jmath}{xi}_n;(p;{geq};7,;n;{geq};4)$ are detected in the stable homotopy groups of spheres, where ${xi}_n;{in};{pi}_{p^nq+pq-2}M$ is obtained in [2]. The new ones are of degree 2(p - 1)($p^n+p+1$) - 4 and 2(p - 1)($p^n+sp^2$ + sp + (s - 1)) - 7 and are represented up to nonzero scalar by $b_0h_0h_n,;b_0h_0h_n ilde{gamma}_s;{ eq};0;{in};Ext^{*,*}_A^(Z_p,;Z_p)$ in the Adams spectral sequence respectively, where $3;{leq};s;<;p-2$.