= Solution
An elliptic <function> is meromorphic and $\Lambda$-periodic. The argument principle on opposite sides of a period parallelogram cancels, so zeros and poles have equal total multiplicity. The <Weierstrass elliptic function>
$$
\wp(z)=z^{-2}+\sum_{0\ne\omega\in\Lambda}[(z-\omega)^{-2}-\omega^{-2}]
$$
is elliptic. Its half-period values $e_i$ are distinct, and comparing poles and leading coefficients proves $\wp'^2=4\prod_i(\wp-e_i)$. The half-period addition formula gives
$$
\wp(z+\omega_1/2)-e_1=\frac{(e_1-e_2)(e_1-e_3)}{\wp(z)-e_1}.
$$
Differentiate and use the same formula at $z=\omega_1/4$ to identify $(e_1-e_2)(e_1-e_3)=(\wp(\omega_1/4)-e_1)^2$, yielding the required <derivative> ratio.
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