  We consider multi-variable sigma function of a genus $g$ hyperelliptic curve
as a function of two group of variables -jacobian variables and parameters of
the curve. In the theta-functional representation of sigma-function, the second
group arises as periods of first and second kind differentials of the curve. We
develop representation of periods in terms of theta-constants. For the first
kind periods, generalizations of Rosenhain type formulae are obtained, whilst
for the second kind periods theta-constant expressions arepresented which are
explicitly related to the fixed co-homology basis.We describe a method of
constructing differentiation operators for hyperelliptic analogues of $\zeta$-
and $\wp$-functions on the parameters of the hyperelliptic curve. To
demonstrate this method, we gave the detailed construction of these operators
in the cases of genus 1 and 2.
