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Previous work by the authors (this journal, \vol{60} (2008), 1009-1044)produced equations that hold on certain loci of the Jacobian of a cyclic$C_{rs}$ curve. A curve of this type generalizes elliptic curves, and theequations in question are given in terms of (Klein's) generalization ofWeierstrass' $\sigma$-function. The key tool is a matrix with entries that arepolynomial in the coordinates of the affine plane model of the curve, thus canbe expressed in terms of $\sigma$ and its derivatives. The key geometric locion the Jacobian of the curve give a stratification of Brill-Noether type. Theresults are of the type of Riemann-Kempf singularity theorem, the methods aregermane to those used by J.D. Fay, who gave vanishing tables for Riemann's$\theta$-function and its derivatives. The main objects we use were developedby several contemporary authors, aside from the classical definitions:meromorphic differentials were expressed in terms of the coordinates mainly byV.M. Buchstaber, J.C. Eilbeck, V.Z. Enolski, D.V. Leykin, and Taylor expansionsfor $\sigma$ in terms of Schur polynomials also contributed by A. Nakayashiki,in terms of Sato's $\tau$-function. Within this framework, following specificresults for $\sigma$-derivatives given by Y. \^Onishi, we arrive at our mainresults, namely statements on the vanishing on given strata of the partialderivatives of $\sigma$ indexed by Young-diagrams subsets that can be workedout in terms of the Weierstrass semigroup of the curve at its point atinfinity. The combinatorial statements hold not only for Jacobians but for thestratification of Sato's infinite-dimensional Grassmann manifold as well.

Jacobi inversion on strata of the Jacobian of the $C_{rs}$ curve $y^r = f(x)$. II