MATHEMATICAL SECTION. 113 
(= Foyltand (_] + _J) 
1 1 
Ri ra aidedte) (ee afte Mie tat st |e) i al ye iad Fare (53) 
1 1 1 
cage oars (A= a, = Ole Sirieiey 9) ay aa wey # a bo (_I@), 
| (ate ved ona) 
1 1 
a ae _|e ea Be ae aay Karey ch aie stp'a) eee tic =o)! (538) 
We easily deduce the symmetrical relations 
2) (Pa) = Yo Hl) (54) 
Je) (_Je), = _P (55) 
This last equation is well known; it appears here, however, as a 
particular case of a more general relation. The quantity ¢. is the 
argument of the @ functions and ie usually denoted 2; 9,() is 
then denoted by x’ ; Schellbach has 5 for Je) and for _ },(>), 
while Hoiiel, in his Recueil de wie has p and p’, respectively. 
Other relations are 
Ph?) deo Of) deo GK gl") Hay 
Se) TC AP cy 
(ge), 9) oh M2) (Poh ua) HM) glo 
iN = SS = —————: * OO EES OO C 
ea oer Cie ek OO 
The following expressions for the nome q can now be given: 
18 mn 
Pa aay ha 2 (_le), 
ey eng (58) 
The first form is simply Jacobi’s definition; the second gives, 
since 
(_Joo), = 2 tan 4 (_] + _Je) (59) 
q = cot? # (_] + _]o) (60) 
This is one of the best formule for computing gq, especially if the 
modulus does not differ much from unity. The third form may be 
