same equations change only in linear in $\wp_{i,j}$-terms $$\begin{aligned}
\begin{split}
\wp_{2222}&=6\wp_{2,2}^2+4\wp_{1,2}-47\lambda_4\wp_{2,2}+92\lambda_4^2-\frac72\lambda_3\\
\wp_{1222}&=6\wp_{2,2}\wp_{1,2}-2\wp_{1,1}-23\lambda_4\wp_{1,2}-6\lambda_3\wp_{2,2}+23\lambda_3\lambda_4+8\lambda_2 \end{split}\end{aligned}$$

Rosenhain modular form representaion of first kind periods
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Rosenhain [@ros851] was the first who introduced $\theta$-functions with characteristics at $g=2$. There are 10 even and 6 odd characteristics in that case. Let us denote each from these characteristics as $$\varepsilon_j = \left[\begin{array}{c} {\varepsilon_j'}^T\\ {\varepsilon''_j}^T \end{array}   \right],
\quad j=1,\ldots 10$$ where $\varepsilon'_j$ and $\varepsilon''_j$ are column 2-vectors with entries equal to 0 or 1.

Rosenhain fixed the hyperelliptic genus two curve in the form $$y^2=x(x-1)(x-a_1)(x-a_2)(x-a_3)$$ and presented without proof expression $$\begin{aligned}
\mathcal{A}^{-1} =\frac{1}{2\pi^2 Q^2 } \left( \begin{array}{rr}   -P\theta_2[\delta_2]&Q\theta_2[\delta_1]\\
 P\theta_1[\delta_2]&-Q\theta_1[\delta_1]  \end{array}    \right) \label{RosenhainFormula}\end{aligned}$$ with $$\begin{aligned}
P&=\theta[\alpha_1]\theta[\alpha_2]\theta[\alpha_3],\quad
Q=\theta[\beta_1]\theta[\beta_2]\theta[\beta_3]\end{aligned}$$ and 6 even characteristics $[\alpha_{1,2,3}], [\beta_{1,2,3}]$ and two odd $[\delta_{1,2}]$ which looks chaotic. One of first proofs can be found in H.Weber [@web859]; these formulae are implemented in Bolza dissertation [@bol885] and [@bol886]. Our derivation of these formulae are based on the [*Second Thomae relation*]{} [@tho870], see [@er08] and [@eil18]. To proceed we give the following definitions.

A triplet of characteristics $[\varepsilon_1]$, $[\varepsilon_2,]$, $[\varepsilon_3]$ is called azygetic if

$$\mathrm{exp}\; \imath \pi  \left\{
 \displaystyle{\sum_{j=1}^3{\varepsilon_j'}^T \varepsilon''_j + \sum_{i=1}^3{\varepsilon_i'}^T
\sum_{i=1}^3\varepsilon''_i } \right\} =- 1$$

A sequence of $2g + 2$ characteristics $[\varepsilon_1],\ldots,[\varepsilon_{2g+2} ] $ 