We differentiate the series definition (4.9) of $\gamma$ to get

$$\frac{\partial\gamma}{\partial\eta_{p}}=\frac{3}{16\pi^{2}}\sum_{(n_{1},n_{2})\in\mathbb{Z}^{2}}\frac{a_{p\bar{q}}(\bar{\eta}_{q}+n_{q})}{|(\eta_{1}+n_{1},\eta_{2}+n_{2},\mu)|_{a}^{5}}.$$

Using the elementary formula for indefinite integrals

$$\int\frac{1}{(s^{2}+t^{2})^{5/2}}ds=\frac{1}{3}\frac{s}{(s^{2}+t^{2})^{3/2}t^{2}}+\frac{2}{3}\frac{1}{t^{4}}(\frac{s}{\sqrt{s^{2}+t^{2}}}\mp 1),$$

we see

$$\int\frac{1}{|(\eta_{1},\eta_{2},\mu)|_{a}^{5}}d\mu=A^{-1/2}\gamma_{\pm},$$

or equivalently

$$\frac{\partial}{\partial\eta_{p}}\left(-\frac{1}{8\pi^{2}}\frac{1}{|(\eta_{1},\eta_{2},\mu)|_{a}^{3}}\right)=\frac{\partial}{\partial\mu}\left(\frac{3a_{p\bar{q}}\bar{\eta}_{q}}{16\pi^{2}A^{1/2}}\gamma_{\pm}\right).$$

Thus after summation

$$\frac{\partial\gamma}{\partial\eta_{p}}=\frac{\partial}{\partial\eta_{p}}\sum_{(n_{1},n_{2})\in\mathbb{Z}^{2}}\left(-\frac{1}{8\pi^{2}}\frac{1}{|(\eta_{1}+n_{1},\eta_{2}+n_{2},\mu)|_{a}^{3}}\right)=\frac{1}{2}\frac{\partial\gamma_{p3}}{\partial\mu}=-\frac{1}{2}\frac{\partial\gamma_{p4}}{\partial\mu}.$$

The ‘morever’ statement follows from summing over the elementary differential relations

$$(a_{p\bar{r}}\bar{\eta}_{r})\frac{\partial\gamma_{\pm}}{\partial\eta_{q}}(\eta_{1},\eta_{2},\mu)=(a_{q\bar{r}}\bar{\eta}_{r})\frac{\partial\gamma_{\pm}}{\partial\eta_{p}}(\eta_{1},\eta_{2},\mu),\quad p,q=1,2.$$

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