the vast amount of recent publications, properties of the $\sigma$-function as a function of the first group of variables were discussed, whilst modular part of variables and relevant objects like $\theta$-constant representations of complete Abelian integrals are considered separately. In this paper we deals with $\sigma$ as a function over both group of variables.

Due to the pure modular part of the $\sigma$-variables, we consider problem of expression of complete integrals of first and second kind in terms of theta-constants. A revival of interest in this classically known material accords to many recent publications reconsidering such problems such as the Schottky problem [@fgs17], the Thomae [@eg06; @ef08; @ekz18] and Weber [@nr17] formulae, and the theory of invariants and its applications [@ksv05; @ef16].

The paper is organized as follows. In the Section 2 we consider the hyperelliptic genus $g$ curve and the complete co-homology basis of $2g$ meromorphic differentials, with $g$ of them chosen as holomorphic ones. We discuss expressions for periods of these differentials in terms of $\theta$-constants with half integer characteristics. Theta-constants representations of periods of holomorphic integrals is known from the Rosenhain memoir [@ros851], where the case of genus two was elaborated. We discus this case and generalize 