is called a [*special fundamental system*]{} if the first $g$ characteristics are odd, the remaining are even and any triple of characteristics in it is azygetic.

(Conjectural Riemann-Jacobi derivative formula) Let $g$ odd $[\varepsilon_1], \ldots,  [\varepsilon_g]$ and $g+2$ even $ [\varepsilon_{g+1}], \ldots, [\varepsilon_{2g+2}]$ characteristics create a special fundamental system. Then the following equality is valid $$\mathrm{Det}\left.  \frac{ \partial( \theta[\varepsilon_1]({v}),\ldots,
            \theta[\varepsilon_g]({v}))  }
        {\partial (v_1,\ldots, v_g)}\right|_{{v}=0}= \pm    \prod_{k=1\ldots g+2}\theta[\varepsilon_{g+k}]({0})
\label{RiemannJacobi}$$

(\[RiemannJacobi\]) proved up to $g=5$ [@fro885], [@igu980], [@fay979]

Jacobi derivative formula for elliptic curve $$\begin{aligned}
\vartheta'_1(0)=\pi \vartheta_2(0)\vartheta_3(0)\vartheta_4(0)\end{aligned}$$

Rosenhain derivative formula for genus two curve is given without proof in the memoir [@ros851], namely, let $[\delta_1]$ and $[\delta_2]$ are any two odd characteristics from all 6 odd, then $$\begin{aligned}
 \theta_1[\delta_1] \theta_2[\delta_2]-\theta_2[\delta_1] \theta_1[\delta_2]
= \pi^2\theta[\gamma_1]\theta[\gamma_2]\theta[\gamma_3]\theta[\gamma_4]
\label{RosenhainDerivative}\end{aligned}$$ where 4 even characteristics $[\gamma_1], \ldots, [\gamma_4]$ are given as $[\gamma_i]=[\delta_1]+[\delta_2]+[\delta_i], \; 3\leq i \leq 6$. There are 15 Rosenhain derivative formulae.

The following geometric interpretation of the special fundamental system can be given in the case of hyperelliptic curve. Consider genus two curve, $$\mathcal{C}:\quad   y^2= (x-e_1) \cdots ( x-e_{6})$$ Denote associated homology basis as $( \mathfrak{a}_1, \mathfrak{a}_2;\mathfrak{b}_1,\mathfrak{b}_2 )$. Denote characteristics of Abelian images of branch points with base point $e_6$ as $ \mathfrak{A}_k$, $k=1,\ldots,6$. These are half-periods 