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

14 YOSHIHIROONISHI

ProofofTheorem3.2.Thebestwaytoexplainthegeneralstepoftheinductionisprobablytodemonstrateonlythecasen=4.Thecaseofn=4isclaimedasfollows.Assumethatu,u1,u2,u3,andu4belongtoι(C).Thenwewanttoprovetheequality σ(u+u1+u2+u3+u4)σ3(u u1)σ3(u u2)σ3(u u3)σ3(u u4)i<jσ3(ui uj)

u(j) v(j)=1