871] A CASE OF COMPLEX MULTIPLICATION WITH IMAGINARY MODULUS, 
or, what is the same thing, 
= i {7 4 - 1 ± Vl4 7 4 + 2}. 
We have 7 8 =1, that is, 7 4 =±1. Considering first the case 7 4 =1, here 
u 2 
and thence 
= ±1, 
1 + 2u ° = 1 4- — , =1+2, = 3 or — 1; 
V y 
moreover, u 8 = v 8 =l. We have thus only the non-elliptic formulae 
dy 
dx 
and 
If however, 7 J = — 1, then 
viz. this is 
1 — y 1 1 — cc- 
dy _ 3 dx 
1 — y' 2 1 — x 2 ’ 
■ a , -satisfied by y = — x, 
by y = 
Sx 4- efi 
1 4- 3« 2 ’ 
! = i(-2±VrT2), 
= | (— 1 ± f /^3) = to, 
if to be an imaginary cube root of unity (to 2 +- co 4- 1 = 0); hence 
u 8 = ( 7 ft>) 4 = — ft). 
Moreover, 
_ 2 v? _ 2u 2 ^ 0 
1 4 = 1 4 , = 1 4~ 2to, 
i) y 
or say, 
= co — to 2 , [= V— 3, if to = £ (— 1 4- i V3)]; 
and we thus have, as in the above-mentioned Note, 
(to — co 2 ) x 4- ft) 2 « 3 
giving 
^ 1 — to 2 (to — to 2 ) x 2 ’ 
(¿y (to — to 2 ) db? 
Vl — y 2 .1 +- coy 2 Vl — « 2 . 1 4- cox 2 ’ 
or, what is the same thing, for the modulus & 2 = —to, we have 
/ ,\ a ® 2 ) sn 0 4-1» 2 sn 3 0 
sn (to - to 2 ) 0 = \ A ——r-; 
1 — ft) 2 (to — to 2 ) sn 2 6 
the values of cn (to — to 2 ) 6 and dn (to — to 2 ) 0 are thence found to be 
, . cn 0(1 — to 2 sn 2 0) 
cn (to — to 2 ) 0 = -7 -r—a ; 
1 — to 2 (to — ft) 2 ) sn 2 0 
and 
i / dn0(l 4-&> 2 sn 2 0) 
dn (to — to-) 0 = A 1 — — n ; 
V ’ 1 - ft) 2 (fO-ft) 2 )sn 2 0’ 
which are the formulae of transformation for the elliptic functions. 
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