given by their characteristics, $[\mathfrak{A}_k]$ with $$\begin{aligned}
\mathfrak{A}_k=
\int_{(e_6,0)}^{(e_k,0)} {u}= \frac12 \tau {\varepsilon'}_k
+ \frac12 {\varepsilon''}_k, \quad k=1,\ldots,6\end{aligned}$$

For the homology basis drawn on the Figure we have

0.7mm

(150.00,80.00) (9.,33.)[(1,0)[12.]{}]{} (9.,33.) (21.,33.) (10.,29.)[(0,0)\[cc\][$e_1$]{}]{} (21.,29.)[(0,0)\[cc\][$e_2$]{}]{} (15.,33.)[(20,30.)]{} (8.,17.)[(0,0)\[cc\][$\mathfrak{ a}_1$]{}]{} (15.,48.)[(1,0)[1.0]{}]{} (32.,33.)[(1,0)[9.]{}]{} (32.,33.) (41.,33.) (33.,29.)[(0,0)\[cc\][$e_3$]{}]{} (42.,29.)[(0,0)\[cc\][$e_4$]{}]{} (37.,33.)[(18.,26.)]{} (30.,19.)[(0,0)\[cc\][$\mathfrak{a}_2$]{}]{} (36.,46.)[(1,0)[1.0]{}]{} (100.,33.00) [(1,0)[33.]{}]{} (100.,33.) (133.,33.) (101.,29.)[(0,0)\[cc\][$e_{5}$]{}]{} (132.,29.)[(0,0)\[cc\][$e_{6}$]{}]{} (59.,58.)[(0,0)\[cc\][$\mathfrak{b}_1$]{}]{} (63.,62.)[(1,0)[1.0]{}]{} (15.,33.00)(15.,62.)(65.,62.) (65.00,62.)(119.00,62.00)(119.00,33.00) (15.,33.00)(15.,5.)(65.,5.) (65.00,5.)(119.00,5.00)(119.00,33.00) (70.,44.)[(0,0)\[cc\][$\mathfrak{b}_2$]{}]{} (74.00,48.)[(1,0)[1.0]{}]{} (37.,33.00)(37.,48.)(76.00,48.) (76.00,48.)(111.00,48.00)(111.00,33.00) (37.,33.00)(37.,19.)(76.00,19.) (76.00,19.)(111.00,19.00)(111.00,33.00)

$$\begin{aligned}
 [{\mathfrak A}_1]= \frac{1}{2}
\left[\begin{array}{cc}1&0\\
                       0&0\end{array}\right],\quad
[{\mathfrak A}_2]= \frac{1}{2}
\left[\begin{array}{cc}1&0\\
                       1&0\end{array}\right],\quad
[{\mathfrak A}_3]= \frac{1}{2}
\left[\begin{array}{cc}0&1\\
                       1&0\end{array}\right],\end{aligned}$$

$$\begin{aligned}
[{\mathfrak A}_4]= \frac{1}{2}
\left[\begin{array}{cc}0&1\\
                       1&1\end{array}\right],\quad
[{\mathfrak A}_5]= \frac{1}{2}
\left[\begin{array}{cc}0&0\\
                       1&1\end{array}\right],\quad
[{\mathfrak A}_6]= \frac{1}{2}
\left[\begin{array}{cc}0&0\\
                       0&0\end{array}\right]\end{aligned}$$

$${\mathfrak{A}}_k= \int_{\infty}^{e_k}{u}=   \frac12 \tau.
{\varepsilon}_k
+ \frac12 {\varepsilon'}_k , \quad [ {\mathfrak{A}}_k ] =
 \left[ \begin{array}{c}
 {\varepsilon}_k^T\\   {\varepsilon'}_k^T \end{array} \right]$$ One can see that the set of characteristics $[\varepsilon_k] = [ \mathfrak{A}_k]$ of Abelian images of branch point contains two odd $[\varepsilon_2]$ and $[\varepsilon_4]$ and remaining four characteristics are even. One can check that the whole set of these $6=2g+2$ characteristics is azygetic and therefore the set constitutes special fundamental system. Hence one can write Rosenhain derivative formula $$\theta_1[\varepsilon_2]\theta_2[\varepsilon_4]-\theta_2[\varepsilon_2]\theta_1[\varepsilon_4]=\pm \pi^2
\theta[\varepsilon_1]\theta[\varepsilon_3]\theta[\varepsilon_5]\theta[\varepsilon_6]$$ In this way we gave geometric interpretation of Rosenhain derivative formula and associated set of characteristics in the case when one from 