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

HYPERELLIPTICFUNCTIONSINGENUSTHREE9

Lemma2.10.Letvbea xedpointinκ 1ι(C)di erentfromanypointofκ 1(O).Thenthefunction

u→σ3(u v)

vanishesoforder2atu=(0,0,0).Precisely,onehas

σ3(u v)=σ2(v)u(3)+(d (u(3))≥3).

Proof.Sinceu visonΘ,wehaveσ(u v)=0.Ifwewriteuas(x1,y1)andvas(x2,y2),2.4(1),2.5and2.7implythat2

x(u).Since

d(u(j) v(j))σ3(u v)σ(u v)2σ3=33 σ23σ x1x2+x2x3+x3x1∞= 1 x3= 1dxthederivativeofthefunctionu→u(j)onκ 1ι(C)bydu(j)du(j)dx

du(j)d(u(1) v(1))=dx