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

HYPERELLIPTICFUNCTIONSINGENUSTHREE11

Definition-Proposition2.13.Letnbeapositiveinteger.Ifu∈κ 1ι(C),thenσ(nu)ψn(u):=

σ2(u)σ2(v)22

Proof.IfweregardutobeavariableonC3,thefunction

u→σ(u+v)σ(u v) 1= 1 x(u) .x(v)

333

33

2(u v)

( u(3)3+···)2σ2(v)2= 1