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

2 YOSHIHIROONISHI

Then(1.1)and(1.2)iseasilyrewittenas

σ(u0+u1+···+un) i<jσ(ui uj)

respectively.

Theauthorrecentlygaveageneralizationoftheformulae(1.3)and(1.4)tothecaseofgenustwoin[13].Ouraimistogiveaquitenaturalgenaralizationof(1.3)and(1.4)andtheresultsin[13]tothecaseofgenusthree(seeTheorem3.2andTheorem4.2).Ourgeneralizationofthefunctioninthelefthandsideof(1.4)isalongalinewhichappearedforacurveofgenustwointhepaper[8]ofD.Grant.AlthoughFay’sfamousformula,thatis(44)inp.33of[6],possiblyrelateswithourgeneralizations,noconnectionsareknown.

Nowwepreparetheminimalfundamentalstoexplainourresults.Letf(x)beamonicpolynomialofxofdegree7.Assumethatf(x)=0hasnomultipleroots.LetCbethehyperellipticcurvede nedbyy2=f(x).ThenCisofgenus3anditisrami edatin nity.Wedenoteby∞theuniquepointatin nity.LetC3bethecoordinatespaceofallvaluesoftheintegrals,withtheirinitialpoints∞,ofthe rstkindwithrespecttothebasisdx/2y,xdx/2y,x2dx/2yofthedi erentialsof rstkind.LetΛ C3bethelatticeoftheirperiods.SoC3/ΛistheJacobianvarietyofC.Wehaveanembeddingι:C →C3/Λde nedby Pdx Px2dxP→(∞2y,∞ x′ ′′ x= .. . (n 1)xσ(u)ny′y′′...y2(x2)′(x2)′′...(x)2(n 1)(yx)′(yx)′′...(yx)(n 1)(x3)′(x3)′′...(x)3(n 1)······...···(n 1) (u), (1.4)

2y,u(2)= (x(u),y(u))

∞xdx2y