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

10 YOSHIHIROONISHI

vanishingordersofu→u(j) v(j)(j=1,2)areequaltoorlargerthanmby(2.3).Furthermoretheexpansion

σ(u v v1 v2)

=σ1( v1 v2)(u(1) v(1))+σ2( v1 v2)(u(2) v(2))+σ3( v1 v2)(u(3) v(3))

+(d (u(1) v(1),u(2) v(2),u(3) v(3))≥2)

showsthatthevanishingorderofu→σ(u v v1 v2)ishigherthanorequaltom.Hencemmustbe1.Ontheotherhand,2.2and(2.3)implythat

σ3(u v)=(u(1) v(1))+(d (u(1) v(1))≥2).

Thusthestatementfollows.

Lemma2.12.Ifuisapointofκ 1ι(C),then

σ3(2u)

33(2u) 22(u)2= 2σ33+3σ32 σ333σ2

σ22 σ22σ(u)

tothefunctionσ(2u)/σ(u)4,bringinguclosetoanypointofκ 1ι(C),weobtainthelefthandsideofthedesiredformula.Herewehaveusedthefactthatu→σ3(2u)doesnotvanish,whichfollowsfrom2.9.Thusthethefunctionσ3(2u)/σ2(u)4isafunctiononι(C),thatisσ3(2(u+ ))

σ2(u)4

foru∈κ 1(C).Lemma2.8(1)statesthisfunctionhasitsonlypoleatu=(0,0,0)moduloΛ.Lemma2.2and2.8(2)givethatitsLaurentexpansionatu=(0,0,0)is

2 13u(3)3

( u(3)3+···)4 (2u(3))+26λ7=2