130 
ON ALGEBRAICAL COUPLES. [22 
The formulae are 
thus become 
A'. 
B'. 
C'. 
D'. 
There is a system 
this is 
E. 
much simpler and not essentially less general, if g — — A. They 
#= & +^J, 
ij=ji = yi + 8j, 
j 2 = A 2 Sf + yj, 
X + AF = + — (7 + AS) (oc + Ay) (cc x + Ayi). 
A 
(Two linear moduli.) 
ij =ji = + Bj, 
f = - A?8i - yj, 
X ± AF = + ^ (7 ± A8) (x + Ay) (iCj + Ayj). 
A 
® 2 = - «+ ¿j, 
0 = - 7® ~ 
>' = yi + Sj, 
j 2 = “ A 2 Si - 7), 
X + AF = + ^ (7 + AS) (¿c + Ay) (x x + Ayj). 
?=-si -Zj, 
ij = - 7 i - Bj, 
ji = ryi + gj, 
i 2 — A 2 Sf + 7), 
X + A F = + ^ (7 + AS ) (# + Ay) (#, + Ay). 
A. 
more general than (A.) having a single linear modulus q(6X + F): 
i 2 = a (i — 6j) + ffiqj, 
ij = a' (i - 6j) + 6 qj, 
ji = 7 (i- 6j) + e qj, 
j 2 = 1 0‘ - Oj) + qj, 
dX+Y=q(6x + y) (Ox 1 + yj) ; 
or, without real loss of generality,