If [1:0:....:0] is an s-fold singularity of a degree $r$ hypersurface $F$ in $\mathbb{P}^n$ then the hypersurface can be written as $F=x_0^{r-s}g_s(x_1,...,x_n) + x_0^{r-s-1}g_{s+1}(x_1,...,x_n) + ... +g_r(x_1,...x_n)$. After we dehomogenize it is known that the initial term $g_s(x_1,...x_n)$ is the tangent cone at the singularity in $\mathbb{C}^n$. My question is known about the higher order terms such as $g_{s+1}(x_1,..,x_n)$? Do they admit a some geometric interpretation?

I know that the common locus of $g_s=g_{s+1}=...=g_{s+h}=0$ give the set of points whose line through the origin has intersection multiplicity s+h+1 with the hypersurface. I would like to find a geometric interpretation of just the hypersurface $g_{s+1}=0$.