Harmonic measures are introduced in Ahlfors as follows.
(Let $\Omega$ be a multiply connected region bounded by an outer contour $C_n$ and $n-1$ inner contours $C_1, \dots , C_{n-1}$.) Suppose now that we solve the Dirichlet problem in $\Omega$ with the boundary values 1 on $C_k$ and 0 on the other contours. The solution is denoted by $\omega_k(z)$, and it is called the harmonic measure of $C_k$ with respect to the region $\Omega$. We have clearly $0 < \omega_k(z) < 1$ in $\Omega$ and
$$\omega_1(z) + \omega_2(z) + \cdots + \omega_n(z) = 1.$$
The contours $C_1, \dots , C_{n-1}$ form a homology basis for the cycles in $\Omega$, homology being understood with respect to an unspecified larger region. The conjugate harmonic differential of $\omega_k$ has periods
$$\alpha_{kj} = \int_{C_j}\frac{\partial \omega_k}{\partial n} ds =\int_{C_j} *d\omega_k$$
along $C_j$.
As an exercise, we are asked to do the following.
Prove that $\alpha_{ij}=\alpha_{ji}$. Hint: Apply Theorem 21, Chap. 4.
Below is Theorem 21, Chap. 4.
Theorem 21. A nonconstant harmonic function has neither a maximum nor a minimum in its region of definition. Consequently, the maximum and the minimum on a closed bounded set $E$ are taken on the boundary of $E$.
Actually, I realized that Theorem 21 in the hint is probably a typo. The solution is quite obvious if we use Theorem 19 instead.
Theorem 19. If $u_1$ and $u_2$ are harmonic in a region $\Omega$, then
$$\int_\gamma u_1 \textrm{ }^\ast du_2 - u_2 \textrm{ }^\ast du_1 = 0$$
for every cycle $\gamma$ which is homologous to zero in $\omega$.
Since $\omega_i(z)$ has value 1 on $C_i$ and 0 on $C_j$ where $j \neq i$, we have
$$\int_{C_1 + C_2 + \cdots + C_n} \omega_i \textrm{ }^\ast d\omega_j - \omega_j \textrm{ }^\ast d\omega_i = \int_{C_i} \textrm{ }^\ast d\omega_j - \int_{C_j} \textrm{ }^\ast d\omega_i = 0, $$
which is what we need to prove.