Using Frobenius-Stickelberger-type relations for hyperelliptic curves (Y. Ônishi, Proc. Edinb. Math. Soc. (2) 48 (2005), 705-742), we provide certain addition formulae for any symmetric power of such curves, which hold on the strata Wk, the pre-images in the Jacobian of the classical Wirtinger varieties. In an appendix, we give similar relations for a trigonal curve y3 = (x - b1)(x - b2)(x - b 3)(x - b4). © Walter de Gruyter Berlin · New York 2008.