1 2 
ER Dae. ee at 
The roots are all real. Let us call them in the ordinary way in 
descending order e,, e,, e,, we then find 
143, 1 1—3A 
Ok SE Cali == Og eck EE e . ° . . s (ib 
Case IV: +1 aoe 
The roots are here, too, all real and run when arranged: 
jot gs Sey eee a 
Case VI: A—=iÀ. 
The roots v, and v, are now conjugate complex. If we follow 
the notation generally assumed, we then write: 
1 SEE ME est) 
oie Sa eae 
_ When reducing the p-functions to the elliptic functions of Jacost 
we make use of the following formulae of reduction: ’) 
onl =O oy =|/ 2 
Me 
2 
! 
VI 
Ess e,—e é,—e 
2 3 1 2 
id nd , PS ==" OS : 
aes €,—e, 
! 2 
=e en*(v) 
pes 6 64's!) = GF + ra " sn3(v) . dn*(v) | 
— 3e, + 2V’(e,'—e,')(e,'— e,') 
Ae, ee =a ae 
+ 3e,'+2//(e,'—e,')(e,'—e,') py Pe v= De! + 4(e,'—e,') (e,'—e,') 
Ay (e,'—e,')(e,'—e,') Ay (e,'—e,')(e,'—e,') 
The expression for §:§ becomes in this way: 
in case Il 
pT Palen es Je, — é,) BL 
k'2 —— 
Beer ee a dn(v) 
En Sn VY p(t 59293) — es ae ot Vee a Mt Via 
in case IV / re ee; KP “iy 
; iD I eT dee PN FE 
er Eig —e,=+Ve,—e, —~ ah Ee 
2 Is 8 1 3 
5 Axes sn(v) 
1) See i. a. M. Krause: Theorie der elliptischen Funktionen (Leipzig, TEUBNER 
(p. í35, 136, 147, 148). 7 
