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For elliptic curves, expressions for the periods of elliptic integrals of thesecond kind in terms of theta-constants, have been known since the middle ofthe 19th century. In this paper we consider the problem of generalizing theseresults to curves of higher genera, in particular to a special class ofalgebraic curves, the so-called $(n,s)$-curves. It is shown that therepresentations required can be obtained by the comparison of two equivalentexpressions for the projective connection, one due to Fay-Wirtinger and theother from Klein-Weierstrass. As a principle example, we consider the case ofthe genus two hyperelliptic curve, and a number of new Thomae andRosenhain-type formulae are obtained. We anticipate that our analysis for thegenus two curve can be extended to higher genera hyperelliptic curves, as wellas to other classes of $(n,s)$ non-hyperelliptic curves.

Periods of second kind differentials of (n,s)-curves