  A geometric algorithm is introduced for finding a symplectic basis of the
first integral homology group of a compact Riemann surface, which is a
$p$-cyclic covering of ${\mathbb C} P^1$ branched over 3 points. The algorithm
yields a previously unknown symplectic basis of the hyperelliptic curve defined
by the affine equation $w^2=z^{2g+1}-1$ for genus $g\geq 2$. We then explicitly
obtain the period matrix of this curve, its entries being elements of the
$(2g+1)$-st cyclotomic field. In the proof, the details of our algorithm play
no significant role.
