Let #(u) and #(u) be the usual functions in the theory of elliptic functions. The following two formulae were found in the nineteenth-century. First one is

DETERMINANTEXPRESSIONSFOR

HYPERELLIPTICFUNCTIONSINGENUSTHREE

YOSHIHIROONISHI

1.Introduction

Letσ(u)and (u)betheusualfunctionsinthetheoryofellipticfunctions.Thefollowingtwoformulaewerefoundinthenineteenth-century.Firstoneis

( 1)n(n 1)/21!2!···n!σ(u0+u1+···+un) i<jσ(ui uj)

σ(u)n2

Althoughthisformulacanbeobtainedbyalimitingprocessfrom(1.1),itwasfoundbefore[7]bythepaperofKiepert[9].

Ifwesety(u)=1 ′ ′′= .. . (n 1) ′′ ′′′... (n)······...··· (n 1) (n)... (u). (2n 3) (1.2)

2y.