the Rosenhain expressions to higher genera hyperelliptic curves. Theta constant representations of second kind periods is known after F.Klein [@klein888], who presented closed formula in terms of derivated even theta-constants for the non-hyperelliptic genus three curve. We re-derive this formula for higher genera hyperelliptic curves.

Section 3 is devoted to a classically known problem, which was resolved in the case of elliptic curves by Frobenius and Stickelberger [@fs882]. The general method of the solution of this problem for a wide class, the so-called $(n, s)$ - curves, has been developed in [@bl08] and represents extensions of the Weierstrass’s method for the derivation of system of differential equations defining the sigma-function. All stages of derivation are given in details and the main result is that the sigma-function is completely defined as the solution of a system of heat conductivity equations in a nonholonomic frame. We also consider there another widely known problem - the description of the dependence of the solutions on initial data. This problem is formulated as a description of the dependence of the integrals of motion, which levels are given as half of the curve parameters, from the remaining half of the parameters. The differential formulae obtained permit 