Weierstrass quarter-period derivative identity ($\frac{{\wp }^{\prime }(z-h_i)}{{\wp }^{\prime }(z)}=-\frac{(\wp (h_i/2)-e_i{)}^2}{(\wp (z)-e_i{)}^2}$)

At $z=h_i/2$, evenness and periodicity give $\wp (z+h_i)=\wp (z)$. The Weierstrass half-period translation formula therefore identifies $(\wp (h_i/2)-e_i{)}^2$ with $(e_i-e_j)(e_i-e_l)$. Differentiating the Weierstrass half-period translation formula proves the identity, interpreted as equality of meromorphic functions, including the apparent singularities of its quotients.